The strong-versus-weak F-regularity conjecture

Let $R$ be a ring of characteristic $p>0$, and let $I^*$ denote the tight closure of an ideal $I$ in $R$. The ring $R$ is strongly $F$-regular when every suitable Frobenius splitting condition holds; the source uses this standard notion.

Strong-versus-weak F-regularity conjecture. $R$ is strongly $F$-regular if and only if

$$I^*=I$$

for all ideals $I$.

A number of special cases are known, and the forward implication follows from the method cited in the source. The converse was presented as an important open problem in tight closure theory.