Hida–Urban non-abelian Leopoldt conjecture for eigenvariety components

Let $\tilde\pi$ be a non-critical cuspidal representation, let $x_{\tilde\pi}\in\cE_{\cU,h}^\bullet$ be the associated point, and let $\cW$ be the weight space. Write $\ell$ for the defect and let $\cV$ be an irreducible component of the eigenvariety passing through the cuspidal point $x_{\refpi}$. Hida–Urban's non-abelian Leopoldt conjecture. The component has dimension

$$\dim(\cV)=\dim(\cW)-\ell.$$

This conjecture controls the dimensions of eigenvariety components and, in particular, predicts that when $\ell=0$, equivalently when $G(\mathbb R)$ admits discrete series, cuspidal families contain a Zariski-dense set of classical points. It is known when $\ell\leq 1$, while the inequality $\dim(\cV)\geq\dim(\cW)-\ell$ is known in general.