The negative-invariant conjecture for nearly Gorenstein normal graded rings

Let $R = \bigoplus_{n\ge 0} R_n$ be a normal graded domain finitely generated over a field $k=R_0$. Let $K_R$ be its canonical module, let $\operatorname{K^{-1}_R}$ be the inverse canonical module, and define

$$b(R)=\min\{n\in\mathbb Z\mid [\operatorname{K^{-1}_R}]_n\ne 0\}.$$

A normal graded ring $R$ is nearly Gorenstein when its canonical trace ideal satisfies $\operatorname{Tr}_R(K_R)=\mathfrak m_R$. If $R$ has an isolated singularity, then:

Negative-invariant conjecture. If $\operatorname{char}(k)=0$ and $b(R)<0$, then $R$ is of strongly F-regular type. If, moreover, $R$ is $\mathbb Q$-Gorenstein, meaning that $r\,\operatorname{cl}(K_R)=0$ in $\operatorname{Cl}(R)$ for some positive integer $r$, then $R$ is log-terminal.

This proposes a stronger singularity-theoretic consequence of the negativity condition on $b(R)$, analogous to the known fact that rational singularities have negative $a$-invariant. The source presents the statement as a conjecture; its resolution status is not specified in the supplied text.