Directional maximum conjecture for logarithmic derivatives of Dirichlet $L$-functions

Let $q$ tend to infinity through the primes. For a prime $q$, let $\chi_0$ denote the principal character modulo $q$, and let $L(s,\chi)$ be the Dirichlet $L$-function associated with a character $\chi$ modulo $q$. For $\theta\in[0,2\pi]$ and $\sigma\in[\frac12,1]$, consider the maximum over nonprincipal characters $\chi\neq\chi_0$. Directional maximum conjecture. As prime $q\to\infty$,

$$\max_{\substack{\chi\neq\chi_0\\ \chi\ (\operatorname{mod} q)}} \operatorname{Re}\left(e^{-\mathrm{i}\theta}\frac{L'}{L}(\sigma,\chi)\right)\sim \frac{\displaystyle\max_{\substack{\chi\neq\chi_0\\ \chi\ (\operatorname{mod} q)}}|L'(\sigma,\chi)|}{\displaystyle\max_{\substack{\chi\neq\chi_0\\ \chi\ (\operatorname{mod} q)}}|L(\sigma,\chi)|},$$

uniformly for all $\theta\in[0,2\pi]$ and all $\sigma\in[\frac12,1]$. This predicts that the largest directional real part of the logarithmic derivative is asymptotic to the ratio of the separate largest derivative and function values; the source motivates it using simultaneous control of prime phases and the belief that central $L$-values do not vanish, but no resolution is given.