The Hurwitz zeta moment conjecture for rational parameters

For a fixed rational number $0 < \alpha \leq 1$, define

$$M_k(T;\alpha)=\int_T^{2T}\left|\zeta\left(\tfrac{1}{2}+it,\alpha\right)\right|^{2k}\,dt.$$

Hurwitz zeta moment conjecture. For every fixed real number $k>0$, there is a constant $c_k(\alpha)$ such that

$$M_k(T;\alpha)\sim c_k(\alpha)T(\log T)^{k^2}$$

as $T\to\infty$. This extends the expected moment growth for the Riemann zeta function to Hurwitz zeta functions with rational parameter; the statement is presented as an expectation, and no resolution is supplied here.