Directional maximum conjecture for the logarithmic derivative of the Riemann zeta function

Fix $\epsilon\in(0,\frac12)$. For $T>0$, consider the interval $[T,2T]$, and let $\zeta(s)$ denote the Riemann zeta function. Directional maximum conjecture. As $T\to\infty$,

$$\max_{T\le t\le 2T}\operatorname{Re}\left(e^{-\mathrm{i}\theta}\frac{\zeta'}{\zeta}(\sigma+\mathrm{i}t)\right)\sim\frac{\displaystyle\max_{T\le t\le 2T}|\zeta'(\sigma+\mathrm{i}t)|}{\displaystyle\max_{T\le t\le 2T}|\zeta(\sigma+\mathrm{i}t)|},$$

uniformly for all $\theta\in[0,2\pi]$ and all $\sigma\in[\frac12+\epsilon,1]$. The source presents this as the zeta-function analogue of the preceding Dirichlet $L$-function conjecture and gives no evidence of a resolution.