The Dirichlet $L$-function analogue of the Hughes–Keating–O'Connell moment conjecture

Let $\kappa\geq 1$ be an integer, let $\chi$ be a primitive Dirichlet character of modulus $q$, and let $\rho=\beta+i\gamma$ range over the non-trivial zeros of the Dirichlet $L$-function associated with $\chi^\kappa$. For $r\in\mathbb{R}$ define

$$\widetilde{J}_r(T)=\sum_{0<\gamma\leq T}|L'(\rho,\chi^\kappa)|^{2r}.$$

Dirichlet $L$-function moment conjecture. For all $r\in\mathbb{C}$ with $\Re(r)>-3/2$, there is a constant $C_{q,r}$ such that

$$\widetilde{J}_r(T)\sim C_{q,r}T(\log T)^{(r+1)^2}.$$

This extends the conjectural moment order for $\zeta'(\rho)$ to derivatives of Dirichlet $L$-functions; the asserted asymptotic remains open.