A remarkable feature of the reduction showing that TQBF (True Quantified Boolean Formulas) is PSPACE-complete is that it actually runs in logspace, i.e. for any $A \in \mathsf{PSPACE}$, $$ A \le_L TQBF. $$ (See for example, comment at the end of section 7.4 in these notes.) Therefore, TQBF is complete with respect to logspace reductions. It is still possible, however, that there are some other standard PSPACE-complete problems for which this is not true. Of course, the problem would reduce in logspace to TQBF, but maybe the reduction the other way would not go through.
Question: If $A$ is PSPACE-complete, must $A$ be complete with respect to logspace reductions? In other words, if $A \in \textsf{PSPACE}$ and $TQBF \le_p A$, then must $TQBF \le_L A$?
Some thoughts:
I tried the generalized geography problem, and it seems to be true in that case. I'm not sure about universality for NFAs.
To prove it, the most straightforward idea would be to show that if $A \le_p B$ and $B \le_L A$, then $A \le_L B$. But this is false for general $A, B$, assuming $\mathsf{L} \ne \mathsf{P}$. In particular, take $A \in \mathsf{P}$ and $B = \varnothing$.