BHS automorphic trianguline subvariety conjecture

Let $X_{\rm tri}^{\mathfrak{X}^p\rm-aut}(\overline{\rho}_p)$ be the rigid subvariety of $X_{\rm tri}^\square(\overline{\rho}_p)$ associated with the tame-level datum $\mathfrak{X}^p$, and let

$$\widetilde X_{\rm tri}^\square(\overline{\rho}_p):=\prod_{v\in S_p}\widetilde X_{\rm tri}^\square(\overline{\rho}_{\tilde v}).$$

BHS automorphic trianguline subvariety conjecture. The rigid subvariety $X_{\rm tri}^{\mathfrak{X}^p\rm-aut}(\overline{\rho}_p)$ does not depend on $\mathfrak{X}^p$ and is isomorphic to $\widetilde X_{\rm tri}^\square(\overline{\rho}_p)$.

The paper recalls this as the main conjecture of BHS and explains that it implies the classical modularity conjectures of BHS. The supplied text does not state whether it has been resolved in this setting.