Central-line maximal logarithmic derivative conjecture for Dirichlet $L$-functions

Let $q$ tend to infinity through the primes. For a prime $q$, let $\chi_0$ be the principal character modulo $q$, and let $L(s,\chi)$ be the Dirichlet $L$-function associated with $\chi$. Write $\log_2 q=\log\log q$. Central-line maximal logarithmic derivative conjecture. There exist constants $A>0$ and $D>\frac12$ such that, as prime $q\to\infty$,

$$\max_{\substack{\chi\neq\chi_0\\ \chi\ (\operatorname{mod} q)}}\operatorname{Re}\left(e^{-\mathrm{i}\theta}\frac{L'}{L}\left(\frac12,\chi\right)\right)\sim A\sqrt{\log q}(\log_2 q)^D,$$

uniformly for all $\theta\in[0,2\pi]$. This conjectures the scale of the largest directional logarithmic derivative at the central point, with the constants constrained only by $A>0$ and $D>\frac12$; no resolution is supplied.