Conjecture on the mean square of the zeta-zero ordinate sum

Let $G(s)$ be the function considered above, and let $T$ tend to infinity. Mean-square conjecture.

$$\int_1^T \left|G\left(\frac{1}{2}+it\right)\right|^2\,dt \asymp T\log^2 T.$$

The preceding theorem gives the unconditional bounds $T\log^2 T\ll\int_1^T|G(\frac12+it)|^2\,dt\ll T\log^2T\sqrt{\log\log T}$; the conjecture predicts removal of the extra factor $\sqrt{\log\log T}$ and is motivated by the expected regularity of the zeta-zero ordinates.