The Halo conjecture for the $\operatorname{GL}_2$ eigencurve

Let $p$ be an odd prime, let $\mathcal{W}$ be the weight space for $\operatorname{GL}_2$, and let $\mathcal{X}$ be an eigencurve with weight map $\operatorname{wt}:\mathcal{X}\to\mathcal{W}$ and slope map $a_p:\mathcal{X}\to\mathbb{G}_m^{\operatorname{rig}}$. For $r\in(0,1)$, write $\mathcal{W}^{>r}=\{|T|>r\}$ and $\mathcal{X}^{>r}=\operatorname{wt}^{-1}(\mathcal{W}^{>r})$. For a point $z$ of $\mathcal{X}$, let $T_{\operatorname{wt}(z)}$ be the weight-space parameter defined by $T_\chi=\chi(\exp(p))-1$. The Halo conjecture. There exists some $r\in(0,1)$ sufficiently close to $1$ such that: (1) $\mathcal{X}^{>r}$ is a disjoint union of countably infinitely many components $\mathcal{X}_1^{>r},\mathcal{X}_2^{>r},\ldots$, and $\operatorname{wt}:\mathcal{X}_n^{>r}\to\mathcal{W}^{>r}$ is finite flat for every $n$; (2) there are rational numbers $\alpha_1\leq\alpha_2\leq\cdots\in\mathbb{Q}_{\geq0}$ tending to infinity such that, for every $z\in\mathcal{X}_n^{>r}$,

$$v(a_p(z))=\alpha_n\,v(T_{\operatorname{wt}(z)});$$

and (3) the sequence $(\alpha_n)$, counted with multiplicities, is a disjoint union of finitely many arithmetic progressions. The conjecture predicts a rigid and eventually patterned geometry for the eigencurve near the boundary of weight space. The source later states that the full Halo conjecture is established, so this conjectural formulation is solved.