The $z$-permanent anti-concentration conjecture

Let $\mathbb{T}$ denote the unit circle, let $X$ be sampled from the complex Gaussian ensemble $\mathcal{G}=\mathcal{N}_{\mathbb{C}}(0,1)^{n\times n}$, and let $\operatorname{Per}_z(X)$ denote the $z$-permanent. $z$-permanent anti-concentration conjecture. There exists a polynomial $p$ such that for every positive integer $n$, every real number $\delta>0$, and every $z\in\mathbb{T}$,

$$\Pr_{X\sim\mathcal{G}}\mathopen{}\mathclose\bgroup\originalleft[ |\operatorname{Per}_z(X)|^2 \geq \frac{\sqrt{n!}}{p(n,1/\delta)}\aftergroup\egroup\originalright] \geq 1-\delta.$$

The source calls this the generalization of the original permanent anti-concentration conjecture. The supplied status evidence says that the case $z=-1$ is proved, while the case $z=1$ is described as a widely believed standard conjecture; consequently the general statement is refuted as a conjecture in its unrestricted form.