A note on mod-$p$ local-global compatibility via Scholze's functor

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.