Cantor-weighted discrete restriction conjecture for the logarithmic phase

Let $\widetilde\nu$ be the Cantor measure pushed forward to $(0,1)$, let $H_{M-1}=\sum_{n=1}^{M-1}n^{-1}$ be the harmonic number, and set $M=\lfloor\sqrt t\rfloor$. Cantor-weighted discrete restriction conjecture. There is an absolute constant $C$ such that, for every $t\ge 2$,

$$\int\Bigl|\sum_{n=1}^{M-1}n^{-1/2+it}e^{-in\theta}\Bigr|^2\,d\widetilde\nu(\theta)\le C H_{M-1}.$$

This is the upper bound on the discrete off-diagonal quantity needed for the Cantor-weighted second-moment conjecture; the supplied discussion says that only a weaker bound is known, so this restriction estimate remains open.