F-purity conjecture for the nearly commuting-matrix ring

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

F-purity conjecture. The quotient ring $R/I$ is $F$-pure for all $n$.

The conjecture concerns the singularities of the algebraic set of nearly commuting matrices in positive characteristic. The source says it can be approached by a subsequent monomial-coefficient conjecture, but gives no resolution of either statement.