The title is the question.


Given a locally compact completely $T\_{4}$ space $X$ (every subspace is $T\_{4}$) and a (Hausdorff) compactification $\overline{X}$ of $X$, is $\overline{X}$ also completely $T\_{4}$?


I have been unable to think of an obvious counterexample, but I suspect that there is one.


The only kind of literature I have been able to find that is related is the following:


[On Compactifications](https://projecteuclid.org/euclid.kjm/1250525055)


Any leads would be appreciated.



Are compactifications of completely $T_{4}$ spaces completely $T_{4}$?
The obvious (to me) counterexamples are $\beta \mathbb{N}$ and $\beta \mathbb{R}$ ( the Čech-Stone compactifications) which are non-completely normal compactifications (classic fact, see Engelking or Gilman and Jerrison's book) of $\mathbb{N}$ and $\mathbb{R}$ resp., which fit your bill (locally compact and even metrisable). 


A [paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1025.8642&rep=rep1&type=pdf) on non-normal subspaces of $\beta \mathbb{N}$ (and hence of $\beta \mathbb{R}$ too as $\beta \mathbb{N}$ embeds in that space naturally).


