The geometric Breuil–Mézard conjecture for potentially crystalline deformation stacks

Let $\lambda$ be a Hodge type, viewed as an element of the character lattice of $G$, and let $\tau$ be a tame inertial type. Let $V(\lambda-\eta_{\mathbb{Q}_p})$ denote the restriction to $\mathcal{G}^\circ$ of the irreducible algebraic representation with highest weight $\lambda-\eta_{\mathbb{Q}_p}$, and let $\operatorname{JH}(\overline{\bar V(\operatorname{DL}^{-1}(\tau))\otimes V(\lambda-\eta_{\mathbb{Q}_p})})$ denote the set of Jordan–Hölder factors of the indicated semisimplified reduction. The geometric Breuil–Mézard conjecture. If $\lambda$ is regular dominant and $\tau$ is a sufficiently generic tame inertial type, then

$$\mathcal{X}^{\operatorname{crys},\lambda,\tau}_{{^{L}\!G},\operatorname{red}}=\bigcup_{\sigma\in\operatorname{JH}(\overline{\bar V(\operatorname{DL}^{-1}(\tau))\otimes V(\lambda-\eta_{\mathbb{Q}_p})})}\mathcal{C}_\sigma.$$

This is the potentially crystalline geometric Breuil–Mézard prediction, generalizing the corresponding result for $\operatorname{GL}_n$; its validity for general tame groups remains open.