Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?
It is known that a 0-dimensional locally compact space which is also paracompact must be strongly 0-dimensional (Engelking, 1989, p. 362). But the answer to a recent question posted here points out that $\omega_1$ is locally compact and pseudocompact but not paracompact, so an approach attempting to use that fact will not answer the present question.
An infinite collection $\mathcal{A}$ of infinite subsets of $\mathbb{N}$ is said to be almost disjoint (AD) if $A\cap B$ is finite whenever $A,B \in \mathcal{A}$ with $A \neq B$. If the family is maximal with respect to this property, then it is called a MAD family.
Given an AD family $\mathcal{A}$, there is a well known way (introduced by Mrowka in "On completely regular spaces", Fund.Math.,1954) to construct a topological space $\Psi(\mathcal{A})$. This space has the following properties:
1) For any AD family, $\Psi(\mathcal{A})$ is 0-dimensional, locally compact and first countable.
2) $\Psi(\mathcal{A})$ is pseudocompact if and only if $\mathcal{A}$ is a MAD family.
Teresawa (in "Spaces N∪R need not be strongly 0-dimensional", Bull. Acad. Polon. Sci. Sér. Sci. Math. Astonom. Phys., 1977) proved that there is a MAD family $\mathcal{A}$ for which $\Psi(\mathcal{A})$ is not strongly 0-dimensional. This provides a counterexample to your question.
