Mod-$p$ local-global compatibility conjecture for unitary groups

Let $F$ be the relevant number field, let $K^p$ be a tame level with split ramification, and let $x\in\mathbb{X}(L)$ be a regular classical point such that $\rho_x$ is residually irreducible and $m(\pi_x)=1$. For each $v\nmid p$, let $\bar{\pi}_{x,v}$ be the associated representation of $U(F_v)$, and set

$$m=\bigotimes_{v\nmid p}\bar{\pi}_{x,v}^{K_v},\qquad b(\bar{\rho}_x)=\operatorname{Hom}_{\mathcal{H}(K^p)}(m,b_x).$$

Mod-$p$ local-global compatibility conjecture. The tautological map

$$\Phi:b(\bar{\rho}_x)\otimes\left(\bigotimes_{v\nmid p}\bar{\pi}_{x,v}^{K_v}\right)\longrightarrow b_x$$

is an isomorphism preserving the actions of $G(\mathbb{Q}_p)$ and $\mathcal{H}(K^p)$. This is proposed as a guiding principle; the source notes that it is known when $S(K^p)=\varnothing$, but otherwise does not establish it.