Slope stability conjecture for ordinary algebraic-geometric \p-adic towers

Let $K_{\infty}/K$ be an algebraic-geometric ordinary $p$-adic Lie tower with no proper constant subextension. Write

$$P(K_n,s)=\prod_{i=1}^{2g_n}(1-\alpha_i(n)s),$$

and normalize $v_q$ by $v_q(q)=1$. The $q$-slopes are the rational numbers $v_q(\alpha_i(n))$, and $r_p(n,\alpha)$ denotes the multiplicity of a fixed rational slope $\alpha\in[0,1]$.

Slope stability conjecture. The following hold: (1) the $q$-slopes are equidistributed in $[0,1]$ as $n\to\infty$; (2) for each fixed rational $\alpha\in[0,1]$, there is a polynomial $R_\alpha(x)\in\mathbb{Q}[x]$ of degree at most ${\rm \dim}(G)$, depending on the tower, such that for all sufficiently large $n$,

$$r_p(n,\alpha)=R_\alpha(p^n);$$

(3) there is a positive integer $n_0$, depending on the tower, such that for all $n>n_0$, the rescaled slopes

$$\{p^nv_q(\alpha_1(n)),\ldots,p^nv_q(\alpha_{2g_n}(n))\}$$

are determined explicitly by their values for $0\leq n\leq n_0$ using finitely many arithmetic progressions.

This conjecture refines $p$-rank and genus stability into predictions for the full slope data of zeta functions. The source presents the three parts as conjectural and gives no resolution status.