The p-stability criterion for strongly F-regular rings

Let $R$ be a strongly $F$-regular ring, and let $\mathfrak{b}_\bullet$ be a $p$-family in $R$. The system has $p$-stabilization $\widetilde{\mathfrak{b}}_e$ in degree $e$.

The p-stability criterion. The system $\mathfrak{b}_\bullet$ is $p$-stable if

$$\widetilde{\mathfrak{b}}_e=\mathfrak{b}_e$$

for all $e$, if and only if there exist an integer $d$ and a surjective map $\varphi\in\mathscr{C}^R_d$ such that

$$\varphi(F_*^d\mathfrak{b}_{e+d})\subseteq\mathfrak{b}_e$$

for all $e$.

The preceding proposition proves the reverse implication for any Noetherian $F$-finite ring, while the conjecture asserts the converse under strong $F$-regularity. The claim concerns when $p$-stability can be detected by a single surjective Cartier-algebra map.