Lamzouri's maximum-order conjecture for Dirichlet L-functions

Let $L(s,\chi)$ be the Dirichlet $L$-function, and let the maximum range over primitive nontrivial Dirichlet characters $\chi$ modulo integers $q\leq Q$. Lamzouri's maximum-order conjecture. As $Q\to\infty$,

$$\max_{q\leq Q}\ \max_{\substack{\chi\neq\chi_0\ \mathrm{primitive}\ (\mathrm{mod}\ q)}}|L(1,\chi)|=(e^\gamma+o(1))\log\log Q,$$

where $\chi_0$ is the trivial character and $\gamma$ is Euler's constant. The conjecture concerns the maximal size of $L(1,\chi)$ among primitive nontrivial characters; the supplied source says that no progress has been made in the direction discussed there.