The characteristic-p absolute-value conjecture for reciprocal zeroes

Let $L(x,y)=\prod_i(1-\beta_i^{(y)}/x)$, where the nonzero $\beta_i^{(y)}$ are the zeroes of $L(x,y)$, and let $\lambda_i^{(y)}:=1/\beta_i^{(y)}$ be the reciprocal zeroes. The characteristic-$p$ absolute-value conjecture. There exists a positive real number $b=b(y)$ such that, if $\delta\geq b$, then there is at most one reciprocal zero of $L(x,y)$, counted without multiplicities, having absolute value $\delta$. The same assertion is intended for all interpolations of $L(s)$ at all places of $\mathbf k$.

This refines the separable-extension formulation by imposing a direct condition on the zeroes. It is stated as an open characteristic-$p$ analogue of the classical simplicity and Riemann-hypothesis phenomena.