F-regularity conjecture for the irreducible polynomial of a generic matrix

Let $X=(x_{ij})$ be an $n\times n$ matrix of indeterminates over a field $K$, and let $\mathcal{P}(X)$ be the irreducible polynomial specified in the source's Definition poly.

F-regularity conjecture for $\mathcal{P}(X)$. The quotient ring

$$K[X]/\mathcal{P}(X)$$

is $F$-regular.

This is one of the source's proposed conjectures about rings arising from the nearly commuting-matrix calculations. The defining polynomial is not included in the supplied context, and no resolution is stated.