Coleman–Mazur–Buzzard–Kilford halo conjecture for eigencurves

Let $p$ be an odd prime, let $\mathscr{W}$ be the rigid analytic space parametrizing continuous characters of $\mathbb{Z}_p^\times$, and write $T(w)=w(\exp(p))-1$ for the $T$-coordinate of a weight $w\in\mathscr{W}$. For $r\in(0,1)$, let $\mathscr{W}_{>r}$ be the rigid analytic subset where $|T|>r$. Fix a tame level, let $\mathscr{Z}$ be the corresponding eigencurve, let $w:\mathscr{Z}\to\mathscr{W}$ be the weight map, let $a_p:\mathscr{Z}\to\mathbb{G}_m$ be the map given by the $U_p$-eigenvalue, and let $\mathscr{Z}_{>r}$ be the preimage of $\mathscr{W}_{>r}$. Coleman–Mazur–Buzzard–Kilford halo conjecture. When $r\in(0,1)$ is sufficiently close to $1^-$, the space $\mathscr{Z}_{>r}$ is a disjoint union of countably infinitely many connected components $Z_1,Z_2,\dotsc$ such that the weight map $w:Z_n\to\mathscr{W}_{>r}$ is finite and flat for each $n$, and there exist nonnegative rational numbers $\alpha_1,\alpha_2,\dotsc\in\mathbb{Q}$, in non-decreasing order and tending to infinity, such that for every $n$ and every point $z\in Z_n$,

$$|a_p(z)|=|T(w(z))|^{\alpha_n}.$$

This conjecture describes the geometry of the eigencurve over the boundary of weight space and is commonly called the halo conjecture. It was stated here as the Coleman–Mazur–Buzzard–Kilford conjecture, following Liu–Wan–Xiao; the supplied material does not establish whether it has been resolved.