Small-height zero abundance conjecture for Dirichlet $L$-functions

Let $N(T,\chi)$ denote the number of zeros of the Dirichlet $L$-function associated with a character $\chi$ having imaginary part in the relevant height range, and let the conductor of $\chi$ be the modulus governing its analytic conductor. Small-height zero abundance conjecture. For every real $T>0$ and every integer $M\ge1$, there is an integer $q_0$ such that every character $\chi$ with conductor at least $q_0$ satisfies $N(T,\chi)\ge M$. This is a qualitative assertion that, for fixed height, the number of zeros tends to infinity with the conductor. It follows from the generalized Riemann hypothesis for Dirichlet $L$-functions by Selberg's estimate, and is known unconditionally in the ranges indicated in the source, but remains open in general.