Bounded digit-sum conjecture for non-classical zero sets of interpolated $L$-functions

Let $k$ be a global function field, let $w$ be any place of $k$, and let $N_w$ denote the non-classical set for the interpolation of $L(\chi,s)$ at $w$; when $w=\infty$, this interpolation is $L(\chi,s)$ itself. The set $N_w$ is known to be closed under multiplication by $q_\chi$. Bounded digit-sum conjecture. The non-classical set $N_w$ consists of elements with bounded sum of $p$-adic digits. This conjecture extrapolates from calculations for several examples and proposes a general restriction on the non-classical zero set; the supplied text gives no resolution.