Zero sets in pseudocompact spaces

Question

Let $X$ be a Tychonoff pseudocompact space and $Z$ be a zero-set in $X$ (i.e., there is a continuous function $f:X \to \mathbb{R}$ such that $x\in Z$ if and only if $f(x)=0$). It is known that these conditions imply that $Z$ is a weakly pseudocompact space (that is, $Z$ is $G_\delta$ dense in some compactification of it). Is it true that $Z$ must actually be pseudocompact? Since the result is usually stated as "zero-sets of pseudocompact spaces are weakly pseudocompact" my guess is that there must be a counterexample, but I've not been able to find it.

Answer 1

The space $\Psi$ provides a counter-example:

It is completely regular, T2, pseudocompact. Its subspace $D$ is a zero-set. Since $D$ is infinite and discrete, it is not pseudocompact.

See the book of Gillman, Jerison, Rings of continuous functions, 5I.