Versal Breuil--Mézard conjecture for $\mathrm{GSp}_4$

Let ${\mathcal S}$ be a set of types, and let $\mathrm{JH}(\overline{\sigma}({\mathcal S}))$ be the union of the Jordan--Hölder factors of the reductions $\overline{\sigma}(\lambda,\tau)$ for $(\lambda+\eta,\tau)\in{\mathcal S}$. Let $\overline{\rho}\in{\mathcal X}_{{\mathrm{Sym}}}(\mathbf F)$ and let $R_{\overline{\rho}}^{\mathrm{alg}}$ be the algebraic versal ring at $\overline{\rho}$. Versal Breuil--Mézard conjecture. For every $\sigma\in\mathrm{JH}(\overline{\sigma}({\mathcal S}))$, there exists an effective cycle ${\mathcal Z}_\sigma(\overline{\rho})\in\mathbf Z[\operatorname{Spec}R_{\overline{\rho}}^{\mathrm{alg}}]$ such that, for all $(\lambda+\eta,\tau)\in{\mathcal S}$,

$${\mathcal Z}_{\lambda,\tau}(\overline{\rho})=\sum_{\sigma\in\mathrm{JH}(\overline{\sigma}(\lambda,\tau))}[\overline{\sigma}(\lambda,\tau):\sigma]{\mathcal Z}_\sigma(\overline{\rho}).$$

This is the versal local formulation obtained by pulling back the geometric conjecture to a versal ring; the source presents it as the corresponding conjecture for $\mathrm{GSp}_4$.