Compatibility map conjecture for the mod-$p$ and $p$-adic constructions

Let $B(\rho_x)^{\circ}$ and $\mathcal{B}_x^{\circ}$ be the lattices in the $p$-adic constructions, let $M^{\circ}$ be the specified lattice in $M=\bigotimes_{v\nmid p}\pi_{x,v}^{K_v}$, and let $b(\bar{\rho}_x)$ and $b_x$ be the mod-$p$ objects. Compatibility map conjecture. There is a $G(\mathbb{Q}_p)$-equivariant map $\Psi$ such that the diagram

$$B(\rho_x)^{\circ}\otimes\mathbb{F}_L\longrightarrow b(\bar{\rho}_x)$$ $$\operatorname{Hom}_{\mathcal{H}(K^p)^{\circ}}(M^{\circ}\otimes\mathbb{F}_L,\mathcal{B}_x^{\circ}\otimes\mathbb{F}_L)\longrightarrow \operatorname{Hom}_{\mathcal{H}(K^p)^{\circ}}(M^{\circ}\otimes\mathbb{F}_L,b_x)$$

commutes, where the right vertical map is composition with $M^{\circ}\otimes\mathbb{F}_L\hookrightarrow m$. The source presents this as an expected compatibility, and does not prove it in general.