Quantum variance conjecture for compact hyperbolic manifolds

Let $\Gamma\backslash\mathbb{H}^n$ be compact, let $f$ be a smooth compactly supported function on this quotient, and let $d\mu_j$ be the measures associated with Laplace eigenfunctions as above. Write $t_j$ for the spectral parameters and

$$\overline{f}=\frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H}^n)}\int_{\Gamma\backslash\mathbb{H}^n}f(z)\,d\mu(z).$$

Quantum variance conjecture. One has

$$\sum_{|t_j|\leq T}\left|\int_{\Gamma\backslash\mathbb{H}^n}f(z)\,d\mu_j(z)-\overline{f}\right|^2=O_f\left(T^{q_n}\right),\qquad q_n=q_n(\Gamma)=1+\epsilon,$$

for every $n\geq 2$. This is described as a weak version of the quantum variance problem. Any upper bound of order $o(T^n)$ would imply quantum ergodicity, but the stated conjecture is far open.