Is Quasi-polynomial time in PSPACE?
Here's a simple argument that shows that QP is not known to be in PSPACE:

Assume $QP \subseteq PSPACE$. Then we have $P \subsetneq QP \subseteq PSPACE$, where the first inclusion is proper by the time hierarchy theorem.

This separates $P$ from $PSPACE$, which is not known to hold, so $QP \subseteq PSPACE$ must also not be known to hold.

Indeed we have that $PSPACE \subseteq QP \Rightarrow PSPACE \subsetneq EXP$, but $QP \nsubseteq PSPACE$ does not separate the two classes by the THT (as stated in the question).