Pearce-Crump's family-averaged Dirichlet ratio moment conjecture

Let $q\not\equiv2\pmod4$, and let $\varphi^*(q)$ denote the number of primitive characters modulo $q$. For each primitive character $\chi$ modulo $q$, write the non-trivial zeros of $L(s,\chi)$ as $\rho=\tfrac12+i\gamma_\chi$. Assume the Generalised Riemann Hypothesis for every $L(s,\chi)$ and that all its zeros are simple. Family-averaged Dirichlet ratio moment conjecture. As $T\to\infty$,

$$\frac{1}{\varphi^*(q)}\sum_{\chi\bmod q}^{*}\;\sum_{0<\gamma_\chi\le T}\left|\frac{L(2\rho,\chi^2)}{L'(\rho,\chi)}\right|^2\sim\frac{\varphi(q)}{q}\,\frac{T}{2\pi},$$

and this asymptotic is expected to hold uniformly for $q\le T^A$, for any fixed $A>0$. The conjecture follows formally by averaging the characterwise prediction, but its asserted uniform asymptotic remains open.