4 problems

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.

Regular-sequence conjecture. The displayed collection is a regular sequence on $R/I$ and hence is part of a system of parameters on $R/J$ and $R/Q$.

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

F-regularity conjecture for nearly commuting matrices. The rings