Inertial local Langlands conjecture for $\operatorname{GL}_n$ with monodromy zero

Let $K$ be a finite extension of $\mathbb Q_p$, let $\tau$ be an inertial type, and let $\widetilde\tau$ be a Frobenius-semisimple Weil–Deligne representation of $W_K$. Inertial local Langlands conjecture. There is a finite-dimensional smooth irreducible $\overline{\mathbb Q}_p$-representation $\sigma(\tau)$ of $\operatorname{GL}_n(\mathcal O_K)$ such that the restriction of $\operatorname{rec}_p^{-1}(\widetilde\tau)^\vee$ to $\operatorname{GL}_n(\mathcal O_K)$ contains $\sigma(\tau)$ if and only if $\widetilde\tau|_{I_K}\sim\tau$ and $N=0$ on $\widetilde\tau$. If $p>n$, then $\sigma(\tau)$ is unique up to isomorphism. This conjectural correspondence is used to formulate the generalized Breuil–Mézard conjecture; the provided text does not state its resolution.