F-regularity conjecture for the commutator ideals

Let $K$ be a field of positive prime characteristic, and let $R$, $\mathcal{I}$, and $J$ be the ring and ideals defined from the commutator matrix in the paper. F-regularity conjecture. The quotient rings

$$R/\mathcal{I}\quad\text{and}\quad R/J$$

are $F$-regular. In particular, they are integral domains. This would strengthen the preceding results that $R/J$ is $F$-pure and reduced in positive characteristic, and would imply strong structural properties for the associated algebraic sets.