Consequences of a distillation algorithm for PSPACE
Theorem 15.3 of the recent "Parameterized Algorithms" textbook by Cygan et al. states the following:

"Let $L, R ⊆ \Sigma^*$ be two languages. If there exists an OR-distillation of L into R, then $L\in coNP / poly$"

So, I think that if there exists an OR-distillation from a PSPACE-complete language $L$ to itself, then $PSPACE \subseteq coNP/poly$, i.e. not only does the polynomial-hierarchy collapse, but also PSPACE collapses with it.