Koyama's mean Lindelöf hypothesis for Rankin–Selberg $L$-functions

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. Then there exists an absolute and effectively computable constant $K>0$ such that, for every $\tau\in\mathbb{R}$ and $\varepsilon>0$,

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

Koyama's mean Lindelöf hypothesis. The displayed bound should hold.

The source explains that this hypothesis concerns the first spectral moment and that the corresponding result is needed in its arguments. It attributes the formulation to Koyama; the status of the stated bound as a conjecture is not resolved by the supplied text.