Weak Generalized Riemann Hypothesis

Let $\chi$ be a Dirichlet character, viewed in the source as a map $\chi:\mathbb{F}_p^*\longrightarrow\mathbb{C}^*$. Let $L(s,\chi)=\sum \chi(n)/n^s$ be its Dirichlet $L$-function, and call a zero nontrivial if it is one of the nontrivial roots considered in the Generalized Riemann Hypothesis.

Weak Generalized Riemann Hypothesis. There exists a constant $\frac{1}{2}>\epsilon\geq 0$ such that all nontrivial roots of $L(s,\chi)$ lie in

$$\frac{1}{2}-\epsilon<\operatorname{Re}(s)<\frac{1}{2}+\epsilon.$$

This is presented as a weaker form of the Generalized Riemann Hypothesis, which places the nontrivial roots on the line $\operatorname{Re}(s)=\frac{1}{2}$. The source does not state a resolution, so the conjecture remains open.