Least nontrivial value conjecture for primitive Dirichlet characters

Let $\chi$ be a Dirichlet character, and let $n_\chi$ be the first natural number such that $\chi(n_\chi)\neq1$. Character least nontrivial value conjecture. For every fixed $\varepsilon>0$, one has

$$n_\chi\ll q^\varepsilon$$

for every primitive Dirichlet character $\chi$ of prime conductor $q$. This is presented as a strengthening of Vinogradov's conjecture in the paper's argument. The supplied material does not state whether this strengthened formulation is known or open.