Eigenvariety and trianguline deformation space component conjecture

Fix a prime $q\neq p$ splitting in $E$, choose $q'$ as in the source, and let $G$ be a unitary group in $d$ variables attached to $E/\mathbb{Q}$ satisfying the stated conditions at infinity and at finite places. Let $\pi$ be an automorphic representation of $G$ with the stated ramification, supercuspidal $\pi_q$, unramified $\pi_p$ with $d$ distinct Satake-parameter eigenvalues, and let $Y_\pi$ be the minimal eigenvariety containing $\pi$. For the associated residual representation $\bar\rho$, classical point $z\in Y_\pi$, and trianguline deformation space $X(\bar\rho,z)$, assume that $\bar\rho$ is absolutely irreducible, the Hodge--Tate weights $k_1<\cdots<k_d$ are pairwise distinct, and $\rho_\pi|_{\mathcal{G}_p}$ is non-critically refined. Eigenvariety component conjecture. The map

$$Y_\pi(\bar\rho)\longrightarrow X(\bar\rho,z)$$

is an inclusion of irreducible components of $X(\bar\rho,z)$. This is a geometric comparison between an eigenvariety and a trianguline deformation space; the source presents the claim as a conjecture and gives no resolution.