Conjecture on moments of the hybrid zero factor derivative

Assume the Riemann Hypothesis. Let $\rho$ range over the non-trivial zeros of $\zeta(s)$, let $N(T)$ count those zeros with $0<\operatorname{Im}(\rho)\leq T$, and let $Z_X(s)$ be the hybrid Euler–Hadamard zero factor. Write $\mathscr{L}=\log(T/(2\pi))$, let $\gamma_0$ be Euler's constant, and suppose $\varepsilon>0$ and $X,T\to\infty$ with $X=O((\log T)^{2-\varepsilon})$. Hybrid zero-factor moment conjecture. For any $k>-3/2$,

$$\frac{1}{N(T)}\sum_{0<\gamma\leq T}|Z_X'(\rho)|^{2k}\sim\frac{G^2(k+2)}{G(2k+3)}(e^{\gamma_0}\log X)^{2k}\left(\frac{\mathscr{L}}{e^{\gamma_0}\log X}\right)^{k(k+2)}.$$

The conjecture is motivated by a random-matrix model for the zeros remaining after removing the short Euler product. The paper proves the case $k=1$ assuming RH and gives evidence for $k=2$, but the full range remains open.