Is every PSPACE-complete problem complete with respect to logspace reductions?
If every PSPACE complete problem is also complete under logspace reduction, then $\mathsf{P\neq PSPACE}$.

To see why, suppose for the purpose of contradiction that the definition of completeness remains unchanged when replacing polynomial time with logspace reductions, and simultaneously $\mathsf{P=PSPACE}$. Since $\mathsf{P=PSPACE}$, every nontrivial language in PSPACE is complete under polynomial time reductions, e.g. undirected connectivity which lies is $\mathsf{L}$. Seeing that we assumed that polynomial time completeneess is equivalent to logspace completeness, undirected conectivity is also PSPACE complete under logspace reductions, which implies $\mathsf{L=PSPACE}$, contradicting the space hierarchy theorem.

This question is probably still open (though I am unaware of any important/strange implications of the negation of your statement). You can find similar discussions about NP-completeness, where it appears the same question is also open.