The square mean Lindelöf hypothesis over the Picard orbifold

Let $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[i])$, let $u_j$ be the relevant Maaß cusp forms with spectral parameters $t_j$, and let $L(s,u_j\otimes u_j)$ be their Rankin–Selberg $L$-functions. Define $\eta\in[0,1]$ by requiring that there exists an absolute and effectively computable constant $K>0$ such that, for every $\tau\in\mathbb{R}$ and $\varepsilon>0$,

$$\sum_{t_j\leq T}\left|\frac{t_j}{\sinh\pi t_j}L\left(\frac12+i\tau,u_j\otimes u_j\right)\right|^2\ll_{\varepsilon}(1+|\tau|)^K T^{3+\eta+\varepsilon}.$$

Square mean Lindelöf hypothesis. The value $\eta=0$ is admissible.

This is a second-moment strengthening of the mean Lindelöf framework and is related through the Watson–Ichino formula to Lindelöf-on-average bounds in the quantum variance problem. The supplied text gives no resolution of the assertion, so it is open.