Wan–Xiao–Zhang slope conjecture for overconvergent p-adic modular forms

Let $p$ be a prime, let $\mathcal{W}$ be the $p$-adic weight space, and let $\mathcal{W}_\eta$ be one of its components. For a weight $\kappa\in\mathcal{W}_\eta$, write $\nu_1(\kappa)\leq\nu_2(\kappa)\leq\dotsb$ for the slopes of the Newton polygon of the Fredholm series $P(\kappa,t)$, and let $w(\kappa)$ be the corresponding weight-space parameter. Wan–Xiao–Zhang slope conjecture. There exists an $r>0$ such that, for each component $\mathcal{W}_\eta$, the Newton polygon of $P(\kappa,t)$ depends only on $v_p(w(\kappa))$ whenever $0<v_p(w(\kappa))<r$, its break-point indices are independent of $\kappa$ on this region, and for every break-point index $i$ one has $\mu(a_{i,\eta})=0$. Moreover, the sequence

$$\left\{\frac{\nu_i(\kappa)}{v_p(w(\kappa))}\right\}$$

is a finite union of arithmetic progressions independent of $\kappa$ for $0<v_p(w(\kappa))<r$ and $\kappa\in\mathcal{W}_\eta$. This conjecture describes the asymptotic slopes of overconvergent $p$-adic modular forms near the boundary of weight space. It is known completely for tame level $N=1$ when $p=2$ or $3$, while the general boundary-slope behavior and the arithmetic-progression assertion remain open.