The non-abelian Leopoldt conjecture for eigenvariety components

Let $G$ be a reductive group, let $K^p\subset G(\mathbb{A}_f^p)$ be a tame level, let $\operatorname{Iw}\subset G(\mathbb{Q}_p)$ be an Iwahori subgroup, and set $K=K^p\operatorname{Iw}$. Let $\mathcal{X}_{G,K}$ and $\mathcal{X}_{G,K}^{\varepsilon}$ be the corresponding eigenvarieties over the weight space $\mathcal{W}$, and let $\ell(G)$ denote the defect of $G$. For a non-critical classical tempered cuspidal eigenpacket $\phi$, write $x_\phi$ for the associated point. The non-abelian Leopoldt conjecture. Any irreducible component $\mathcal{I}\subset\mathcal{X}_{G,K}$, or $\mathcal{X}_{G,K}^{\varepsilon}$, through $x_\phi$ has dimension

$$\dim \mathcal{W}-\ell(G).$$

This predicts that the defect measures the codimension of cuspidal eigenvariety components in the weight space, extending the usual Leopoldt philosophy to non-abelian eigenvarieties. The conjecture is attributed to Hida and Urban; the supplied text gives no evidence of a general resolution.