Hirose–Watanabe–Yoshida's Gorenstein criterion for standard graded rings

Let $R$ be a standard graded strongly $F$-regular ring, with $R_0$ an $F$-finite field of characteristic $p>0$. Let $\mathfrak{m}$ be the unique homogeneous maximal ideal of $R$. Hirose–Watanabe–Yoshida's conjecture. The equality

$$\operatorname{fpt}(\mathfrak{m})=-a(R)$$

holds if and only if $R$ is Gorenstein.

The conjecture characterizes Gorenstein standard graded strongly $F$-regular rings using the $F$-pure threshold of their homogeneous maximal ideal. The paper proves it under the additional hypothesis that the anti-canonical cover of $R$ is Noetherian.