  We remove the semisimple condition in the mod-$p$ local-global compatibility
result of arXiv:2106.10674. Namely, assuming flatness of
$\pi_{\mathfrak{m}}^\vee$ and
$\overline{\sigma}_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ being
multiplicity free, we prove that
$H^{n-1}_{\text{\'et}}(\mathbb{P}_{\mathbb{C}_p}^{n-1},
\mathcal{F}_{\pi_{\mathfrak{m}}[\mathfrak{m}]})$ determines
$\overline{\sigma}_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$
uniquely. We give remarks on our assumptions at the end of this note.
