The density conjecture for uncyclic matrices

Let $q\geqslant 2$ be a finite field size and let $n\geqslant 1$. Write $\operatorname{unc}(n,q)$ for the number of uncyclic matrices in $\operatorname{M}(n,q)$, where an uncyclic matrix is one for which every primary component of the underlying $\mathbb{F}_q[X]$-module $\mathbb{F}_q^n$ is non-cyclic. The density conjecture. The proportion of uncyclic matrices satisfies

$$\frac{\operatorname{unc}(n,q)}{q^{n^2}} \leqslant \frac{1}{q}\left(\frac{1}{q}+\frac{1}{2q^2}\right)^n.$$

The conjecture gives an upper bound for the density of uncyclic matrices, equivalently a lower bound for the density of $f$-cyclic matrices relevant to the Meat-axe algorithm. The paper confirms the bound for dimensions $n\leqslant 37$, while the general assertion for all finite field sizes and dimensions is left open.