Conjecture on extreme Dirichlet L-values in cosets

Let $q>1$, let $\mathcal{X}$ be the group of Dirichlet characters modulo $q$, and let $H\leq\mathcal{X}$ be a subgroup satisfying $\#H\gg q^\varepsilon$. For any coset representative $\chi_0\in\mathcal{X}/H$, the coset is $\chi_0H$. Coset extreme-value conjecture. Every such coset contains an extreme value over the whole group, namely

$$\max\limits_{\chi\in\chi_0H}L(\chi,1/2)=\exp\left((\sqrt{1/2}+o(1))\sqrt{\log q\log\log q}\right).$$

This extends the Gonek-Hughes-Farmer prediction from the full character group to every sufficiently large subgroup coset. The source presents it as a proposed conjecture and gives no resolution status.