Does graph connectivity being NP-complete imply NL=P?
If NP completeness of connectivity implies $\mathsf{NL}=\mathsf{P}=\mathsf{NP}$, then $\mathsf{P}\neq\mathsf{PSPACE}$. To see why, suppose $\mathsf{P}=\mathsf{PSPACE}$, then connectivity (and any other non trivial language in P) is NP complete, thus $\mathsf{NL}=\mathsf{P}=\mathsf{PSPACE}$ (by our original assumption), which contradicts the space hierarchy theorem. 

This means such implication is not yet known. For completeness, I'll add that a negative answer would imply the existence of an NP complete problem which is not complete under logspace reductions (if all NP complete problems were also complete under logspace reductions, then NP completeness of connectivity implies $\mathsf{NL}=\mathsf{NP})$.