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.
Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?