Displaced discrete-moment conjecture for the Riemann zeta function

Let $N(T)$ denote the number of non-trivial zeros $\tfrac12+{\mathrm{i}}\gamma_n$ with $0<\gamma_n\leq T$, and set

$$L=\frac{1}{2\pi}\log\frac{T}{2\pi}.$$

For $k$ with $\operatorname{Re}(k)>-1/2$, let $a(k)$ be the arithmetic factor defined by

$$a(k)=\prod_{p\ \operatorname{prime}}\left(1-\frac{1}{p}\right)^{k^2}\sum_{m=0}^{\infty}\left(\frac{\Gamma(m+k)}{m!\,\Gamma(k)}\right)^2p^{-m},$$

and let $F_k$ be the function specified in the paper's theorem on displaced characteristic-polynomial moments. Displaced discrete-moment conjecture. As $T\to\infty$,

$$\frac{1}{N(T)}\sum_{0<\gamma_n\leq T}\left|\zeta\left(\tfrac12+{\mathrm{i}}\left(\gamma_n+\frac{\alpha}{L}\right)\right)\right|^{2k}\sim\frac{G^2(k+1)}{G(2k+1)}a(k)F_k(2\pi\alpha)\left(\log\frac{T}{2\pi}\right)^{k^2},$$

uniformly in $\alpha$ for $|\alpha|\leq L$. Here $G$ is the Barnes $G$-function. This conjecture unifies the continuous moment conjecture and the discrete moments of the zeta derivative through the limiting cases $\alpha\to\infty$ and $\alpha\to0$; the general assertion remains open.