The unimodular density conjecture for monic rectangular polynomial matrices

For a positive integer $m$ and integers $k<n$, define

$$M_{n,k}({\mathbb F}_q[x];m):=\left\{x^m{I_k \brack \bf{0}}+\sum_{i=0}^{m-1}x^iA_i: A_i\in M_{n,k}({\mathbb F}_q)\text{ for }0\leq i\leq m-1\right\}.$$

Here, a polynomial matrix is unimodular if its maximal minors generate the unit ideal in ${\mathbb F}_q[x]$, and let $\delta_q(n,k)=\prod_{i=1}^{k}(1-q^{i-n})$.

Unimodular density conjecture. The probability that a uniformly random element of $M_{n,k}({\mathbb F}_q[x];m)$ is unimodular is given by

$$\delta_q(n,k).$$

The preceding finite-field enumeration establishes this density for the corresponding constant-coefficient family, and the conjecture asks whether the same probability persists for every positive degree parameter $m$.