Does the Labor Theory of Value hold in the long term in competitive markets? Most criticisms of the Labor Theory of Value I have come across go something like this quote from Carl Menger:

There is no necessary and direct connection between the value of a
good and whether, or in what quantities, labor and other goods of
higher order were applied to its production. A non-economic good (a
quantity of timber in a virgin forest, for example) does not attain
value for men since large quantities of labor or other economic goods
were not applied to its production. Whether a diamond was found
accidentally or was obtained from a diamond pit with the employment of
a thousand days of labor is completely irrelevant for its value.

This is perfectly obvious and Marx even mentions this:

A commodity, such as iron, corn, or a diamond, is therefore, so far as
it is a material thing, a use value, something useful. This property
of a commodity is independent of the amount of labour required to
appropriate its useful qualities.

But putting aside the natural resource example for a moment, in the long term in a competitive market isn't it the case that if there is a situation where someone is able to sell something for more than the cost of the labor that went into it, someone else should be able to step in and sell that thing for a lower price? Eventually prices will go down until they are stopped by another constraint. That constraint being the cost of all the labor required to produce the product.
In situations where this doesn't happen isn't it an indication of some anti-competitive force?
Monopoly of a natural resource, monopoly of a technology through a patent, externalities that result in a small number of large companies, cooperation between large companies to set prices, etc...
So is it fair to say that in the long term in competitive markets the labor theory of value holds?
If the theory is &quot;at any point in time a commodity is worth as much as the cost of labor that went into it&quot; then it is obviously wrong and I can't imagine any classical economist would have believed it.
But if the theory is &quot;over time the cost of a good will tend towards the cost of the amount of labor that went into it assuming the market is competitive&quot; then it still seems plausible.
No it does not for several reasons. Labor theory of value (LTV) central tenant is that there is some intrinsic component to value determined objectively by labor  (see Brue and Grant History of Economic Thought  7ed). This implies that there exist some unit of labor in which every value could be measured. In addition, it means that the value ($V$) is some sort of function labor $L$ i.e. $V=f(L)$. As stated by Adam Smith in Wealth of Nations who first developed the LTV (and as mentioned in the comments as well):

&quot;The value of any commodity ... is equal to the quantity of labour which it enables him to purchase or command. Labour, therefore, is the real measure of the exchangeable value of all commodities.

Hence, at the core of labor theory of value is proposition that all value can be somehow objectively measured by labor, and that it should depend on labor.
Well there are several problems with that proposition:

What about things that require great labor inputs but are worthless (which is situation that can and empirically does arise even in competitive markets as preferences and tastes change and something that was once desired might be trash today)?

The LTV actually has no good answer for that. Adam Smith tried to
answer this objection by distinguishing between value in use and
value in exchange. As Smith further writes in his Wealth of Nations:

The word value, it is to be observed, has two different meanings, and sometimes expresses the utility of some particular object, and sometimes the power of purchasing other goods which the possession of that object conveys. The one may be called &quot;value in use&quot;; the other, &quot;value in exchange.&quot; The things which have the greatest value in use have frequently little or no value in exchange; and, on the contrary, those which have the greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it will purchase scarce anything; scarce anything can be had in exchange for it. A diamond, on the contrary, has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it.

This is basically the argument that Marx repeats (although he had different name for it), as mentioned above (indeed I am always surprised that Marx is often quoted regarding LTV given he did not do virtually any original work on the LTV and just took it from Ricardo and Smith).
The problem with this is that it is already quite scientifically undesirable as it is essentially a cop-out. Labor is the source of value, well, except in cases where it isn't because the object is not useful. So already LTV from the get go admits it is not a theory that can explain all value, which is scientifically undesirable as we want to generalize as much as possible. Now, if we would not have alternative theories of value this could be forgiven but given we do have them and they can explain more this in itself is argument for ditching LTV (e.g. we can safely ditch Newtons theory of gravity in favor of relativity as the latter is more general, but even though theory of relativity can't explain everything we cannot ditch because there is no other competing theory that can).
With regards to your main question, already this point 1 proves that in competitive markets LTV does not need to hold as it is completely valid outcome under perfect competition that value of produced good will be 0 (or indeed even negative like with oil futures).

How to measure value in labor (the problem of transformation)?

One of the central tenets of labor theory of value as mentioned
above is that labor must be some sort of real measure of value of
all commodities. However, due to the problem mentioned in point 1 it is also necessary to argue that value of labor will not correspond exactly to the  price.
Hence, this gives raise to the question how to actually establish some correspondence between prices and labor which is also known as the problem of transformation (see Samuelson; 1971). This was in fact a problem that Ricardo devoted considerable amount of time to solve, although he was ultimately unsuccessful.
Samuelson argues there is no general solution to the problem, and although there are some scholars claiming to solve it  there are no generally accepted solutions.
In regards to your question, assuming competitive market does not provide solution to the above problem and without the above being possible it is very hard to sustain claim that value can be somehow objectively measured by labor.

Empirically value seems to be orthogonal on labor inputs and this hold regardless of how competitive industry is.

There simply does not seem to be any relationship between value and labor inputs. Relative prices fluctuate widely even when labor inputs stay same. This can occur both in theoretical models of perfect competition (e.g. consider two perfectly sectors two goods model with some idiosyncratic demand shocks), and frequently occurs in reality.
Now even if there would be a discrepancy between price and the value of labor that went into producing the product, if there is any objective value determined by labor then you would at the very least expect relative prices to depend on relative labor inputs but they do not and there is no indication that they start doing so in competitive industries. Hence $V \perp L$.
This also connects to the previous point, but for LTV to hold it is simply not enough to just show that something that was produced with labor is happens to have value - the value of it should depend on labor input and ultimately be measurable.  What even more it should be the  measuring stick of the value. But this is clearly inconsistent with observation that relative prices change and fluctuate widely even when labor inputs do not.