Random permanent zero-probability conjecture over finite fields

Let $p$ be an odd prime and let $q=p^m$ be a fixed power. Let $A$ be a uniformly random $n\times n$ matrix with entries in the finite field $\mathbb{F}_q$.

Random permanent zero-probability conjecture.

$$\Pr[\operatorname{per} A=0]=\frac{1}{q}+o(1).$$

This conjecture asserts that, for fixed odd-characteristic finite fields, the permanent of a random matrix is zero with asymptotic probability $1/q$, as would be expected if the permanent behaved like a random polynomial. The statement concerns the asymptotic regime $n\to\infty$; the source provides no evidence that it has been resolved.