The many-rank matrix-sum equidistribution conjecture

Let $\operatorname{Mat}^{n\times n}_q$ be the additive group of $n\times n$ matrices over $\mathbb F_q$. For integers $r_1,\ldots,r_k,t$ with $0\le r_i,t\le n$, let $\mathcal N^{n\times n}_q(r_1,\ldots,r_k;t)$ denote the number of $k$-tuples of matrices of respective ranks $r_1,\ldots,r_k$ whose sum is a fixed matrix of rank $t$, and let $\mathcal R^{n\times n}_q(r)$ denote the number of matrices of rank $r$. Many-rank matrix-sum equidistribution conjecture. For every $\epsilon>0$ there exists $K(\epsilon)$ such that, if $k\ge K(\epsilon)$ and $\epsilon n\le r_1,\ldots,r_k\le n$, then

$$\mathcal N^{n\times n}_q(r_1,\ldots,r_k;t)=\operatorname{O}(1)\,\frac{\mathcal R^{n\times n}_q(r_1)\cdots\mathcal R^{n\times n}_q(r_k)}{q^{n^2}}.$$

The source introduces this as the second of two conjectural generalizations of results in the paper; no resolution status is supplied.