Hughes–Keating–O'Connell conjecture for discrete moments of zeta'

Let $\zeta(s)$ be the Riemann zeta function, assume the Riemann Hypothesis, and let $N(T)$ count its non-trivial zeros $\rho=\tfrac12+i\gamma$ with $0<\gamma\leq T$. Let $G(z)$ be the Barnes $G$-function, and let $a(k)$ be the arithmetic factor defined by

$$a(k)=\prod_{p\ \operatorname{prime}}\left(1-\frac1p\right)^{k^2}\sum_{m=0}^{\infty}\left(\frac{\Gamma(m+k)}{m!\,\Gamma(k)}\right)^2p^{-m}.$$

Hughes–Keating–O'Connell conjecture. For any fixed $k$ with $\Re(k)>-3/2$,

$$\frac1{N(T)}\sum_{0<\gamma\leq T}\left|\zeta'\left(\tfrac12+i\gamma\right)\right|^{2k}\sim a(k)\frac{G^2(k+2)}{G(2k+3)}\left(\log\frac{T}{2\pi}\right)^{k(k+2)}.$$

This predicts discrete absolute moments of $\zeta'$ using the corresponding characteristic-polynomial calculation, with the same arithmetic factor as in the continuous moment conjecture; the source gives no resolution.