Is there any plausible complexity/crypto hypothesis that rules out the possibility that polynomial size circuits have subexponential-size (i.e. $2^{O(n^\epsilon)}$ with $\epsilon<1$) bounded-depth ($d = O(1)$) circuits?
We know that every function computable by a $\mathsf{NC^1}$ circuit can be computed by a size $2^{O(n^\epsilon)}$ depth $d$ circuit (using AND, OR, and NOT gates, unbounded fan-in) (for every $0 <\epsilon$ there is a $d$ and $d$ can be taken to be $O(1/\epsilon)$).
The question is:
is there a reason that would make the existence of such circuits for general polynomial size circuits unlikely?