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

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

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.