Asymptotic uniformity conjecture for permanents over finite fields

Let $q$ be a prime power such that $Fbb{F}_{q}$ has odd characteristic. For each positive integer $n$, let $A$ be a uniformly random matrix in $Fbb{F}_{q}^{n\times n}$, and write

$$\operatorname{per}(A)=\sum_{\pi\in S_n}\prod_{i=1}^n a_{i,\pi(i)}.$$

Asymptotic uniformity conjecture. For every $x\inFbb{F}_{q}$,

$$\lim_{n\to\infty}\Pr[\operatorname{per}(A)=x]=\frac{1}{q}.$$

The conjecture predicts that, unlike the determinant, the permanent of a uniformly random matrix over a finite field of odd characteristic becomes uniformly distributed as the matrix size tends to infinity. The source presents this as an open conjecture; characteristic two is excluded because the determinant and permanent coincide there.