Permanent anti-concentration conjecture for complex Gaussian matrices

Let $W$ be an $n\times n$ matrix with independent entries distributed as $N(0,1)_{\mathbb C}$, and let $p$ be a polynomial in its arguments. Permanent anti-concentration conjecture (PACC). There exists a polynomial $p$ such that for all $n$ and $\delta>0$,

$$\Pr\left(\left|\operatorname{per}(W)\right|<\frac{\sqrt{n!}}{p(n,1/\delta)}\right)<\delta.$$

This is the standard anti-concentration conjecture for the permanent and is used to control the probability that the permanent of a complex Gaussian matrix is unusually close to zero. Under this conjecture, the paper obtains that only $O_{\epsilon}(\log n)$ zeros of $\operatorname{per}(W+zJ)$ lie within a ball of radius $(1-\epsilon)\sqrt{1/n}$ around the origin, implying that the bulk of the zeros lies at scale $\sqrt{1/n}$; the conjecture itself is not resolved here.