Nilpotence conjecture for solutions of generic matrix commutator equations

Let $n\geq 2$, and let $R$ be a commutative ring with unity satisfying a condition of the form that the integer $\phi(n,\alpha)!$ is not a zero-divisor. Let $\widetilde{R}$ denote the ring over which the generic matrix is defined, let $A\in M_n(\widetilde{R})$ be generic, and let $X=[x_{i,j}]\in M_n(\widetilde{R})$ be a solution of

$$XA-AX=X^{\alpha}.$$

Nilpotence conjecture. One has

$$AX-XA=X^{\alpha}=0_n,$$

and, for every $(i,j)$,

$${x_{i,j}}^{(\alpha-1)n+1}=0.$$

The preceding computations establish this behavior for $n=2$ and $2\leq \alpha\leq 4$, while the stated factorial condition is only specified through an unspecified function $\phi(n,\alpha)$; the general assertion is presented as a conjecture motivated by numerical Gröbner-basis experiments.