What is the latest thinking on the economic calculation problem? Eric Posner and Glen Weyl write in the epilogue of Radical Markets (2018) that they “remain confident that for at least a few generations markets – Radical Markets, that is – will remain the best method of large-scale social organization”.

Are there indications that alternative calculation methods may challenge this thesis?

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CLARIFICATION:

Typically an alternative calculation method, more efficient than markets, is considered to be possible through technology: parallel processing and more recently artificial intelligence:


  The market process … may be considered as a computing device of the pre-electronic age.


Oskar Lange (1967): The Computer and the Market

I wondered what the current discourse was on the topic beyond what is available on Wikipedia.
Depends what exactly is the authors definition of “radical markets”. A predominant consensus in the economic field is that a mixed economy which relies predominantly on market form of organization but also has government stepping in correcting major market failures and some macroeconomic management is economic system that delivers the greatest amount of material welfare to greatest number of people. 

I can’t see much challenge to this consensus within the profession but I would hesitate to make predictions about future - just few years before Einstein developed theory of relativity Lord Kelvin stated that: “There is nothing new to be discovered in physics now, All that remains is more and more precise measurement." Although, the consensus is quite strong and entrenched so I would also not say that the estimate of authors of at least few generations is wrong.

Wether that’s the best is matter of personal morals. Some radical ascetic may prefer a system where everyone lives on some agrarian subsistence level. If our objective would be to just bring  everyone down to the subsistence level then markets would of course not be the best form of organization. This last paragraph might look silly to you but I am just highlighting that there is no moral consensus on what the objective of society should be, and this objective is also what determines what’s the best system, and that this is ultimately question that transcends economics.



Edit: In response to your new question in your clarification about using computer technology, the following statement:


  Typically an alternative calculation method, more efficient than markets, is considered to be possible through technology: parallel processing and more recently artificial intelligence


is not considered correct in economics, in fact this whole statement puzzles me because it presents itself as a fact while it is not true that in the wider economic literature technology, as you defined it here as parallel processing or artificial intelligence, is considered an efficient alternative or even more efficient method of economic calculation.

The fundamental problems that markets help to solve are aggregating information and providing incentives. However, when we talk about information we often talk about information which is not easy or even possible to objectively verify. For example, important information that markets aggregate is the information on consumer preferences. No matter of computing power will help you with that if people dont reveal their real preferences through lets say questionnaires because they either might suffer from social desirability bias, people might have incentives to mislead or not even themselves fully understand the preferences they have. 

Moreover, market mechanism through prices actually provides incentives for people to act in accordance with the information the markets aggregate. If the prices for apples increases due changes in consumer preferences then producers do not get just the information about that in an aggregate and simply understandable way in the form of price but they are also provided with an incentive to act accordingly and expand their output. 

Parallel processing and machine learning does not help with solving the underlying  problem of economic calculation in any way. Parallel processing and machine learning algorithms can certainly help governments to conduct more efficient policy by helping to model and estimate its effect and getting more efficient feedback on the response of economy to government action. But they do not help with the fundamental problem of economic calculation at all, and frankly no amount of computing power probably will as the problem of economic calculation is not a problem of lack of computing power but a problem of eliminating information asymmetry between people and 'central planners'  and giving people incentives to provide accurate information and act on it.