On mod $p$ local-global compatibility for $\mathrm{GL}_n(\mathbf{Q}_p)$ in the ordinary case

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation $\overline{r}:\mathrm{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ is upper-triangular and satisfies certain genericity conditions at a place $w$ above $p$, and that every subquotient of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ of dimension $>2$ is Fontaine--Laffaille generic. In this paper, we show that the isomorphism class of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ is determined by $\mathrm{GL}_n(F_w)$-action on a space of mod $p$ algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to $\overline{r}$, assuming a weight elimination result which is a theorem of Bao V. Le Hung in his forthcoming paper~\cite{LeH}. In particular, we show that the wildly ramified part of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ is determined by the action of Jacobi sum operators (seen as elements of $\mathbf{F}_p[\mathrm{GL}_n(\mathbf{F}_p)]$) on this space.