Conjectural moments of the truncated Hadamard factor

Let $Z_X(s)$ be the truncated Hadamard factor, let $X,T\to\infty$ with $X=O((\log T)^{2-\epsilon})$, and let $k>-1/2$ be fixed. Let $G$ be the Barnes $G$-function and let $\gamma$ be Euler's constant. Truncated Hadamard moment conjecture.

$$\frac1T\int_T^{2T}\left|Z_X\left(\frac12+{\rm i}t\right)\right|^{2k}{\rm d}t \sim \frac{G^2(k+1)}{G(2k+1)}\left(\frac{\log T}{e^{\gamma}\log X}\right)^{k^2}.$$

This is obtained from the random-matrix model for the nearby zeta zeros and supplies the Hadamard-factor contribution to the hybrid moment formula. It is conjectural and is not proved in the source.