Automorphic component conjecture for trianguline varieties

Let $\mathfrak{X}^p$ be an irreducible component of $\mathfrak{X}^\square_{\bar{\rho}^p}$. Let $X$ be an irreducible component of $X_{\rm tri}^\square(\bar{\rho}_p)$. By definition, $X$ is $\mathfrak{X}^p$-automorphic if

$$\iota(X)\times\mathfrak{X}^p\times\mathbb{U}^g$$

is an irreducible component of $X_p(\bar{\rho})$. Automorphic component conjecture. The component $X$ is $\mathfrak{X}^p$-automorphic if and only if

$$X\cap U_{\rm tri}^\square(\bar{\rho}_p)^{\rm reg}$$

contains a crystalline point. The forward implication is known from the density theorem; the conjecture concerns the converse and can equivalently characterize the irreducible components of $X_p(\bar{\rho})$ in this way.