Generalized Breuil–Mézard conjecture for $\operatorname{GL}_n$

Let $K$ be a finite extension of $\mathbb Q_p$, let ${\bar{r}}:G_K\to\operatorname{GL}_n(\mathbb F)$ be continuous, and let $R^\square_{{\bar{r}},\lambda,\tau}$ be the potentially crystalline deformation ring of Hodge type $\lambda$ and inertial type $\tau$. If $(L_{\lambda,\tau}\otimes_{\mathcal O}\mathbb F)^{\operatorname{ss}}\cong\bigoplus_a F_a^{n_a}$, where $a$ ranges over Serre weights, then Generalized Breuil–Mézard conjecture. There exist integers $\mu_a({\bar{r}})$ depending only on ${\bar{r}}$ and $a$ such that

$$e(R^\square_{{\bar{r}},\lambda,\tau}/\pi)=\sum_a n_a\mu_a({\bar{r}}).$$

This is the higher-dimensional analogue of the Breuil–Mézard conjecture, expressing deformation-ring multiplicities through Serre-weight data.