Barvinok's concentration conjecture for the Frobenius permanent estimator

Let $A$) be an $n\times n$ matrix with entries in $\{0,1\}$. For a matrix $A$, define $M(A)$ by

$$M(A)_{ij}=\rho_{ij}A_{ij},$$

where the $\rho_{ij}$ are chosen independently from the Gaussian distribution on $\mathcal{A}_d$. Define $M(\mathds 1)$ similarly by $M(\mathds 1)_{ij}=\rho_{ij}\delta_{ij}$. Let $\operatorname{perm} A$ denote the permanent of $A$, and let $\|\cdot\|$ denote the Frobenius norm. Barvinok's concentration conjecture. There is a sequence of constants $\gamma_d$ with

$$\lim_{d\to\infty}\gamma_d=1,$$

such that, for every $\epsilon>0$,

$$\lim_{n\to\infty}\Pr\left[(\gamma_d+\epsilon)^{-n}\operatorname{perm} A\leq\frac{\|M(A)\|^2}{\|M(\mathds 1)\|^2}\leq(\gamma_d+\epsilon)^n\operatorname{perm} A\right]=1.$$

This conjecture asserts asymptotic concentration of the Frobenius estimator around the permanent, up to a factor that becomes mild as $d$ grows. The paper states that its results do not address the conjecture directly, so its resolution is not established here.