Does $P \neq NP$ imply $NP \neq PSPACE$?
We know that $\mathrm{L}\subseteq\mathrm{NL}\subseteq\mathrm{P}\subseteq\mathrm{NP}\subseteq\mathrm{PSPACE}$ and at least one of those inclusions is strict, since $\mathrm{L}\subsetneq\mathrm{PSPACE}$ by the space hierarchy theorem. As far as I'm aware, any nonempty subset of the inclusions being strict would be consistent with our current state of knowledge so, in particular, it is not known that $\mathrm{P}\neq\mathrm{NP}$ implies $\mathrm{NP}\neq\mathrm{PSPACE}$.