Cantor-weighted Hurwitz zeta alpha-second-moment conjecture

Let $\nu$ be the Cantor measure used in the paper and let $\widetilde\nu$ be its pushforward to $(0,1)$. Let $\zeta(s,\alpha)$ denote the Hurwitz zeta function. Cantor-weighted Hurwitz alpha-second-moment conjecture. As $t\to\infty$,

$$\int \bigl|\zeta(\tfrac12+it,\alpha)\bigr|^2\,d\widetilde\nu(\alpha)=\tfrac12\log t+O(1).$$

The conjecture is intended to provide a Cantor-measure analogue of known Lebesgue $\alpha$-average second-moment formulas and would imply the Lindelöf hypothesis for the associated function $L$; it is open in the supplied text.