Pointwise upper-bound conjecture for the logarithmic derivative of the Riemann zeta function

Fix $\sigma\in(\frac12,1)$, and let $\zeta(s)$ denote the Riemann zeta function. Pointwise upper-bound conjecture. For all sufficiently large $t$,

$$\left|\frac{\zeta'}{\zeta}(\sigma+\mathrm{i}t)\right|\ll_{\sigma}(\log t)^{1-\sigma}(\log_2 t)^{1-\sigma}.$$

The source lists this as a conjectural refinement of conditional upper bounds for logarithmic derivatives; no proof or disproof is stated in the supplied text.