Spectral halo conjecture for Hilbert modular eigenvarieties

Let $I$ index the relevant embeddings, let $\rX_D^{>r}$ be the boundary region of the eigenvariety, let $\rW^{>r}$ be the corresponding weight-space region, let $w$ be the weight map, and let $a_i(x)$ and $T_{i,w(x)}$ denote the quantities associated to the $i$-th component as in the construction. When $r\in (0,1)$ is sufficiently close to $1^-$, there should exist, for every $i\in I$, a sequence of rational numbers

$$\Sigma_i=\\{\alpha_{i,0}\leq \alpha_{i,1}\leq \dots \\}.$$

Writing $\Sigma:=\prod\limits_{i\in I}\Sigma_i$, the eigenvariety $\mathcal{X}_D^{>r}$ should be a disjoint union

$$\mathcal{X}_D^{>r}=\bigsqcup_{\alpha=(\alpha_i)_{i\in I}\in \Sigma}\mathcal{X}_\alpha$$

of possibly empty rigid analytic spaces finite over $\mathcal{W}^{>r}$ via $w$, such that for every $\alpha\in\Sigma$ and each closed point $x\in\mathcal{X}_\alpha(\mathbb{C}_p)$,

$$v_p(a_i(x))=(p-1)v_p(T_{i,w(x)})\cdot \alpha_i$$

for all $i\in I$. Moreover, each sequence $\Sigma_i$ should be a disjoint union of finitely many arithmetic progressions, counted with multiplicities. This conjecture predicts a precise spectral-halo decomposition and valuation pattern near the boundary of the Hilbert modular eigenvariety, extending the Coleman–Mazur–Buzzard–Kilford picture; the supplied text gives no resolution status.