L-function degree and slope stability conjecture for algebraic-geometric $\mathbb{Z}_p$-towers

Let $L(\chi_n,s)$ be the $L$-function of a primitive character of order $p^n$, let $\ell(n)$ be its degree, let $\ell_\alpha(n)$ count the slopes equal to a fixed rational number $\alpha\in[0,\infty)$, and let $v_q$ be the normalized $q$-adic valuation. L-function stability conjecture. If the $\mathbb{Z}_p$-tower comes from algebraic geometry, then:

1. There are constants $a,b$ with $a>0$ such that $\ell(n)=ap^n+b$ for all sufficiently large $n$.
2. For every fixed rational $\alpha\in[0,\infty)$, $\ell_\alpha(n)$ is constant for all sufficiently large $n$.
3. The $q$-slopes of $L(\chi_n,s)$ are equidistributed in $[0,1]$ as $n\to\infty$.
4. There is a positive integer $n_0$, depending on the tower, such that for all $n>n_0$, the rescaled $q$-slopes are explicitly determined by their values for $0\leq n\leq n_0$ using finitely many arithmetic progressions.

This is the $L$-function version of the paper’s main conjecture and is stated to be equivalent to the zeta-function formulation; the supplied text gives no resolution status.