The characteristic-p Generalized Simplicity Conjecture

Let $\overline{k}$ be a fixed algebraic closure of $k$, let $k^{\rm sep}$ be its separable closure, and let $G=\operatorname{Gal}(k^{\rm sep}/k)$. Let $\rho:G\to\mathbf C_\infty^*$ be a character factoring through a finite abelian extension. Let $G_1$ be the Galois group of the maximal constant-field extension of $\mathbf K$ over $\mathbf K$; the orbit of $\rho$ under $G_1$ is its Galois packet. The characteristic-$p$ Generalized Simplicity Conjecture. (1) If $L(s)=L(\rho,s)$, with $s=(x,y)$, then for fixed $y$, almost all zeroes of $L(x,y)$ are simple. (2) If $\rho$ and $\rho'$ belong to distinct Galois packets and $y\in\mathbb Z_p$, then there is a number $c=c(y)$ such that the absolute values of the zeroes of $L(\rho,x,y)$ and $L(\rho',x,y)$ greater than $c$ are distinct.

This is the characteristic-$p$ counterpart of the classical simplicity conjecture. It asserts eventual simplicity within a character's $L$-series and eventual separation of zero absolute values between distinct Galois packets. The paper leaves it as an open conjecture and mentions an analogous $v$-adic version.