In one sense, $P$ vs. $PSPACE$ is the "easiest" first step to showing $P \neq NP$... and this is one you hear often about. In a different sense, you could take $L$ at one end and then $PH$ at the other. Is anything known about this question, whether $L \stackrel{?}{=} PH$?
There is this separation guaranteed by $L \neq PSPACE$, and so the weakest unanswered questions are when you circumvent the time- and space-hierarchy theorems by either asking to strengthen the lower side into a time-bound class (giving $P \stackrel{?}{=} PSPACE$), or by weakening the harder side into a time-bound class (giving $L \stackrel{?}{=} PH$).
It seems that I only ever hear about the first question though, and not the latter. Besides the question of the actual relation between the two classes, is there a reason why one question is so much more popular than the other?