It would be nice to show that $P=L$ implies $NP=NL$. Or, $NP=NL$ implies $UP=UL$. Or maybe, $⊕P = ⊕L$ implies $PP = PL$.
Are there any known connections between the problems: P vs L, UP vs UL, NP vs NL, ZPL vs ZPP, RL vs RP, BPP vs BPL, PP vs PL, or ⊕P vs ⊕L?
I just started to learn about probabilistic space complexity classes so for all I know there are known results or maybe this hasn't even been investigated thoroughly.
An incomplete list of results (correct me if any of them are wrong):
(1) $L \subseteq UL \subseteq NL$
(2) $L \subseteq ZPL \subseteq RL \subseteq BPL \subseteq PL$
(3) $RL \subseteq NL \subseteq PL \cap ⊕L$
(4) $PL \cup ⊕L \subseteq NC^2 \subseteq P$
One connection: if any of these problems have a positive answer, then we get $NC^2 = P$, $PSPACE = EXPTIME$, and $AL \neq AP$ where it is well known that $AL = P$ and $AP = PSPACE$.
For example, suppose that we have $PP=PL$. Then, we get $PL = NC^2 = P = PP$. And, $NC^2 = P$ implies $PSPACE = EXPTIME$. Further, we can't have $AL = AP$. Otherwise, we would get $P = PSPACE = EXPTIME$ which violates the time hierarchy theorem.
Any corrections, ideas, thoughts, results, or references are appreciated. Thank you for reading! :)