Hughes–Pearce-Crump complex discrete moment conjecture for zeta'

Let $\zeta(s)$ be the Riemann zeta function, and assume the Riemann Hypothesis, so that its non-trivial zeros are $\rho=\tfrac12+i\gamma$ with $\gamma$ real. For $T>0$, let $N(T)$ denote the number of such zeros with $0<\gamma\leq T$, with

$$N(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}+O(\log T).$$

Hughes–Pearce-Crump conjecture. For $\Re(k)>-3$,

$$\frac{1}{N(T)}\sum_{0<\gamma\leq T}\zeta'\left(\tfrac12+i\gamma\right)^k\sim\frac{1}{\Gamma(k+2)}\left(\log\frac{T}{2\pi}\right)^k$$

as $T\to\infty$, where $\Gamma(z)$ is the Gamma function. This conjecture predicts the complex discrete moments of the derivative of the zeta function from random-matrix and hybrid-model calculations; the source gives no resolution.