The dual F-signature conjecture for F-injectivity

Let $(R, \mathfrak{m}, k)$ be a reduced Cohen–Macaulay $F$-finite local ring of characteristic $p>0$, and let $\omega_R$ be a canonical module. The dual F-signature conjecture. $R$ is $F$-injective if and only if the $F$-surjective ratio $r(\omega_R)$ is positive.

The $F$-surjective ratio generalizes the $F$-splitting ratio, whose positivity characterizes $F$-purity. This conjecture proposes the analogous characterization of $F$-injectivity using the canonical module.