What if an $L$-complete problem has $NC^1$ circuits? More generally, what evidence is there against $NC^1=L$?

Question

What if an $L$-complete problem has $NC^1$ circuits? More generally, what evidence is there against $NC^1=L$?

Answer 1

If an $L$-complete problem (under first-order projection, or $NC^0$-reduction) has $NC^1$ circuit, then we have $L=NC^1$.

The $NC^1$ circuit for an arbitrary language $\mathbb{L}\in L$ is obtained by first projecting (in case of first-order projection) or doing a constant-depth transformation (in case of $NC^0$-reduction) to obtain an instance of the $L$-complete problem. Then, plugin the $NC^1$ circuit for the complete problem to solve the original problem overall.