Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family

Let $q$ tend to infinity through odd primes, let $T=T(q)$, and put $\ell=\log(qT/2π)$. Suppose that $\ell^s=o(T)$ for some fixed $s>1$ and that \[ λ_{\rm bw}:=\min\left\{1, \liminf_{\substack{q\to\infty\\q\ \mathrm{prime}}} \frac{\log(q-1)}{\ell}\right\}>0. \] For the unweighted family of the $q-2$