Pointwise upper-bound conjecture for logarithmic derivatives of Dirichlet $L$-functions

Fix $\sigma\in(\frac12,1)$. Let $q$ be a sufficiently large integer, let $\chi$ be a primitive character modulo $q$, and let $L(s,\chi)$ be its Dirichlet $L$-function. Pointwise upper-bound conjecture. For all sufficiently large integers $q$ and all primitive characters $\chi\ (\operatorname{mod} q)$,

$$\left|\frac{L'}{L}(\sigma,\chi)\right|\ll_{\sigma}(\log q)^{1-\sigma}(\log_2 q)^{1-\sigma}.$$

The source presents this as the Dirichlet $L$-function counterpart of the preceding zeta bound and does not give a resolution in the supplied text.