Geometric Breuil--Mézard conjecture for the Emerton--Gee stack of $\mathrm{GSp}_4$

Let ${\mathcal X}_{{\mathrm{Sym}},K,\mathrm{red}}$ be the reduced Emerton--Gee stack, and let ${\mathcal Z}_{\lambda,\tau}$ be the $4f$-dimensional cycle attached to a regular Hodge type $\lambda$ and a mildly generic tame inertial type $\tau$. For each Serre weight $\sigma$, seek a $4f$-dimensional cycle ${\mathcal Z}_\sigma$ in this stack. Geometric Breuil--Mézard conjecture.

$${\mathcal Z}_{\lambda,\tau}=\sum_\sigma[\overline{\sigma(\tau)\otimes V(\lambda-\eta)}:\sigma]{\mathcal Z}_\sigma.$$

The source formulates this stack-theoretic version alongside the deformation-ring version and states that the relevant geometric Breuil--Mézard results are proved under genericity assumptions.