F-regularity conjecture for nearly commuting matrix components

Let $X$ and $Y$ be $n\times n$ matrices of indeterminates over a field $K$, let $R=K[X,Y]$, and let $I$ be generated by the off-diagonal entries $u_{ij}$ of $XY-YX$. Let $P=\operatorname{Rad}(J)$, where $J$ is generated by all entries of $XY-YX$, and let $Q$ be the other minimal prime of $I$.

F-regularity conjecture for nearly commuting matrices. The rings

$$R/P,\qquad R/Q,\qquad R/(P+Q)$$

are $F$-regular.

The conjecture is motivated by computer-algebra experiments and is stated to hold when $n=2$. The source explains that, under a linkage reduction, it suffices to establish the $F$-regularity of $R/(P+Q)$, while the general case remains unresolved there.