Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$.

What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\rfloor\}$?

Is there a known distribution at a chosen dyadic interval?

My guess is it is at least $\frac1{p(n)}$ at a fixed polynomial whose degree is independent of $n$.