Conjecture on log-canonical thresholds and a-invariants

Let $(R,\mathfrak{m},K)$ be a standard graded normal $\mathbb{Q}$-Gorenstein algebra over a field $K$ of characteristic zero. Let $d=\dim(R)$ and let $X=\operatorname{Spec} R$. Suppose that $Y$ is log-canonical.

Conjecture on log-canonical thresholds and a-invariants. 1.

$$\operatorname{lct}(X)\leqslant -a_d(R).$$

2. If $R$ is Gorenstein, then

$$\operatorname{lct}(X)=-a_d(R).$$

This conjecture is motivated by the open problem of whether log-canonical pairs are precisely pairs of dense $F$-pure type, and by difficulties involving Cohen–Macaulayness and reduction of the $a$-invariant to positive characteristic. Its resolution status is not specified in the supplied text.