  Let $(R,\mathfrak{m},K)$ be an $F$-finite Noetherian local ring which has a
canonical ideal $I \subsetneq R$. We prove that if $R$ is $S_2$ and
$H^{d-1}_{\mathfrak{m}}(R/I)$ is a simple $R\{F\}$-module, then $R$ is a
strongly $F$-regular ring. In particular, under these assumptions, $R$ is a
Cohen-Macaulay normal domain.
