F-regularity conjecture for partial commutator ideals

Let $X$ and $Y$ be $n\times n$ matrices of indeterminates over a field $K$, let $R=K[X,Y]$, and let $u_{ij}$ be the off-diagonal entries of $XY-YX$. For a subset

$$Z\subseteq\{u_{ij}\mid 1\leq i\neq j\leq n\}$$

of cardinality at most $n^2-n-1$, let $\mathcal{I}_Z$ be the ideal of $R$ generated by the elements of $Z$.

F-regularity conjecture for partial commutator ideals. The quotient $R/\mathcal{I}_Z$ is $F$-regular. In particular, $\mathcal{I}_Z$ is a prime ideal.

This predicts strong singularity properties for every proper partial collection of the off-diagonal commutator equations. The source gives no verification or resolution for the general values of $n$ and $Z$.