Geometric Breuil--Mézard conjecture for $\mathrm{GSp}_4$ deformation rings

Let $G_K$ be the absolute Galois group of $K$, let $\overline{\rho}:G_K\rightarrow\mathrm{GSp}_4(\mathbf{F})$ be continuous, and let $R_{\overline{\rho}}^\square$ be its framed deformation ring. For each Serre weight $\sigma$, let $Z_\sigma$ be a cycle of dimension $4f+11$ in $\operatorname{Spec}R_{\overline{\rho}}^\square/\varpi$. For a regular Hodge type $\lambda$ and a mildly generic tame inertial type $\tau$, write $Z(R_{\overline{\rho}}^{\lambda,\tau}/\varpi)$ for the corresponding cycle and $[\overline{\sigma(\tau)\otimes V(\lambda-\eta)}:\sigma]$ for the Jordan--Hölder multiplicity. Geometric Breuil--Mézard conjecture.

$$Z(R_{\overline{\rho}}^{\lambda,\tau}/\varpi)=\sum_\sigma[\overline{\sigma(\tau)\otimes V(\lambda-\eta)}:\sigma]Z_\sigma.$$

The source states this conjecture for the tamely potentially crystalline case and says that it is proved under appropriate genericity assumptions.