The generic nonsymmetric matrix equation conjecture for cubic systems

Let $K$ be the base field, let $n=3$, and let $B,C,D$ be generic $3\times 3$ matrices. Consider Eq., the nonsymmetric generic matrix equation in the unknown matrix $X\in\mathcal{M}_3(\overline{K})$. The generic cubic system conjecture. The solution count and Galois group of Eq. are

$$(56,S_{56}).$$

For $n=3$, numerical specializations over $\mathbb{Q}$ exhibit a degree-$56$ polynomial with Galois group $S_{56}$, motivating this conjecture; the asserted generic result is not established in the supplied text.