A Breuil–Mézard conjecture for discrete series extended types

Let $K$ be the local field and let $G_K$ be its absolute Galois group. Let $\mathbb{F}$ be the coefficient field, let $\Gamma$ be the relevant Grothendieck group, and let $\overline{\rho}$ be a continuous two-dimensional representation of $G_K$ over $\mathbb{F}$. For a discrete series inertial type $\texttt{t}$, Hodge–Tate type $w$, character $\psi$ lifting $\omega^{-1}\det\overline{\rho}$ and compatible with $\texttt{t}$ and $w$, and extended type $\texttt{t}^+$ compatible with $(\texttt{t},\psi)$, write $R^{\square,\psi}(w,\texttt{t}^+,\overline{\rho})$ for the corresponding potentially semi-stable deformation ring, and let $\sigma_{\mathcal{G}}(\texttt{t})$ and $\sigma_w$ be the associated representations. Breuil–Mézard conjecture for extended types. There exists a positive linear form $\mu_{\overline{\rho}}$ on $\mathcal{R}(\Gamma)$ with values in $\mathbb{Z}$ such that, for every such tuple, one can choose $\sigma_{\mathcal{G}}(\texttt{t})$ with

$$e(R^{\square,\psi}(w,\texttt{t}^+,\overline{\rho})/\pi)=\mu_{\overline{\rho}}([\overline{\sigma_{\mathcal{G}}}(\texttt{t})\otimes\overline{\sigma_w}]).$$

If $\overline{\rho}$ is not a twist of an extension of the trivial character by the cyclotomic character, then the multiplicities for conjugate extended types agree:

$$e(R^{\square,\psi}(w,\texttt{t}^+,\overline{\rho})/\pi)=e(R^{\square,\psi}(w,\texttt{t}^-,\overline{\rho})/\pi),$$

where $\texttt{t}^-$ is conjugate to $\texttt{t}^+$. This proposes extending the usual Breuil–Mézard conjecture to cases not yet known in the paper; the displayed multiplicity formula is intended to relate deformation-ring special fibres to mod-$p$ representation-theoretic data, while the equality for conjugate types addresses the exceptional ambiguity between the two extensions.