Mixed zeta-ratio moment conjecture

Let $a,k,\beta\in(0,\infty)$ and let $\sigma\in(\frac{1}{2},1]$ be fixed. Mixed zeta-ratio moment conjecture. There is a constant ${\mathcal D}(\sigma,a,k,\beta)$, depending only on $a,k,\beta$ and $\sigma$, such that, as $T\to\infty$,

$$\int_T^{2T}\left|\frac{\zeta(\frac{1}{2}+{\rm i}t)}{|\zeta(\sigma+{\rm i}at)|^\beta}\right|^{2k}{\rm d}t\sim {\mathcal D}(\sigma,a,k,\beta)\,T\log^{k^2}T.$$

This extends the paper's second-moment results and is motivated by the conjectured asymptotics for moments of $\zeta(\frac{1}{2}+{\rm i}t)$; the source does not provide a resolution of this general statement.