  A conjecture of Hirose, Watanabe, and Yoshida offers a characterization of
when a standard graded strongly $F$-regular ring is Gorenstein, in terms of an
$F$-pure threshold. We prove this conjecture under the additional hypothesis
that the anti-canonical cover of the ring is Noetherian. Moreover, under this
hypothesis on the anti-canonical cover, we give a similar criterion for when a
normal $F$-pure (resp. log canonical) singularity is quasi-Gorenstein, in terms
of an $F$-pure (resp. log canonical) threshold.
