Hirose–Watanabe–Yoshida conjecture on the F-pure threshold and the a-invariant

Let $k$ be an $F$-finite field and let $S$ be a standard graded algebra over $k$, with homogeneous maximal ideal $\mathfrak{m}$. Assume that $S$ is strongly $F$-regular. The $F$-pure threshold of $S$ with respect to $\mathfrak{m}$ is denoted by $\operatorname{fpt}_{\mathfrak{m}}(S)$. For a $d$-dimensional graded domain, its a-invariant is

$$\operatorname{a}(S)=\min\\{n\in\mathbb{Z}\mid [H^d_{\mathfrak{m}}(S)]_{-n}\neq 0\\}.$$

Hirose–Watanabe–Yoshida conjecture. One always has

$$\operatorname{fpt}_{\mathfrak{m}}(S)\leq -\operatorname{a}(S),$$

and, moreover, $S$ is Gorenstein if and only if

$$\operatorname{fpt}_{\mathfrak{m}}(S)=-\operatorname{a}(S).$$

The conjecture proposes that the F-pure threshold detects the Gorenstein property of strongly F-regular standard graded algebras through equality with the negative a-invariant. The supplied text gives no evidence resolving either assertion.