High-probability rank growth conjecture for generalized Moore matrices

Let $\bm{M} \in \mathbb{F}_{q^m}^{k \times n}$ be a randomly chosen matrix with full $\mathbb{F}_{q^m}$-rank such that each block $\bm{M}^{(i)}$, for $i=1,\dots,\ell$, has full column rank over $\mathbb{F}_q$. Let $\bm{a} \in \mathbb{F}_{q^m}^{\ell}$ consist of randomly chosen representatives of distinct nontrivial conjugacy classes of $\mathbb{F}_{q^m}$, and fix $j \in \{1,\dots,n-k\}$. High-probability rank growth conjecture. The iterated operator $\Gamma_{\bm{a}}^j$ satisfies

$$\operatorname{rk}_{q^m}(\Gamma_{\bm{a}}^j(\bm{M}))=\min((j+1)k,n)$$

with high probability. This claim is analogous to an assumption used for Overbeck's distinguisher and contrasts with the deterministic rank growth of generalized linearized Reed–Solomon codes; establishing the probability of this behavior for random full-rank matrices is the relevant issue.