The local-global compatibility conjecture for spectral p-adic Jacquet-Langlands

Let $E\subset\overline{\mathbb{Q}_p}$ be a finite extension of $\mathbb{Q}_p$, and let $\rho:G_{\mathbb{Q}}\to\operatorname{GL}_2(E)$ be an odd continuous absolutely irreducible representation unramified outside finitely many places. Write $\rho_l$ for its restriction to $G_{\mathbb{Q}_l}$, let $\pi_l$ be the smooth representation of $\operatorname{GL}_2(\mathbb{Q}_l)=D^\times(\mathbb{Q}_l)$ corresponding to $\rho_l$ for $l\ne p$, and let $J(\rho_p)$ be the representation of $D^\times(\mathbb{Q}_p)$ obtained from the local p-adic Jacquet-Langlands correspondence. Let $\mathcal{A}_{E}^{v_p}$ denote the global space introduced above. Local-global compatibility conjecture. There is an isomorphism of $D^\times(\mathbb{Q}_p)$-representations

$$\operatorname{Hom}_{D^\times(\mathbb{A}_f^{(p)})}\left(\bigotimes_{l\ne p}\pi_l,\mathcal{A}_{E}^{v_p}\right)\cong J(\rho_p).$$

This conjecture predicts compatibility between the global spectral $p$-adic Jacquet-Langlands construction and Scholze's local correspondence. The paper derives consequences for locally algebraic vectors assuming the conjecture; the source does not state a resolution.