Gonek-Hughes-Farmer conjecture for maximum Dirichlet L-values

Let $q>1$, let $\mathcal{X}$ be the group of Dirichlet characters modulo $q$, and let $L(\chi,s)$ denote the associated Dirichlet $L$-function. Gonek-Hughes-Farmer conjecture. As $q$ tends to infinity,

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

This conjecture predicts the extreme central value among characters modulo $q$, motivated by random-matrix and random-Euler-product heuristics. The source cites Gonek, Hughes and Farmer for the conjecture and records a substantially weaker lower bound, but gives no resolution status.