The higher-moment conjecture for the Hurwitz zeta function

Let $k\in\mathbb{N}$ and let $0<\alpha\leqslant 1$ be irrational. Write $\zeta(s,\alpha)$ for the Hurwitz zeta function and let $d$ denote the algebraic degree of an algebraic number $\alpha$; for transcendental $\alpha$, let $\mu(\alpha)$ denote its irrationality exponent. Higher-moment conjecture. If $\alpha$ is algebraic of degree $d\geqslant k$, or if $\alpha$ is transcendental with $\mu(\alpha)=2$, then

$$\int_T^{2T}|\zeta(\tfrac12+it,\alpha)|^{2k}\,dt\sim k!\,T(\log T)^k$$

as $T\to\infty$. This predicts the moments expected from diagonal contributions and, in particular, the moments of a complex Gaussian with variance $\log T$. The source notes that the conjecture is known after averaging over the shift parameter in the case $k=2$, while the unaveraged assertion remains open.