Fourth moment conjecture for real-valued Maaß newforms

Let $q$ and $\chi$ be fixed, and let $g\in\mathcal{B}_0^{\ast}(q,\chi)$ be a real-valued newform with spectral parameter $t_g$. Equip $\Gamma_0(q)\backslash\mathbb{H}$ with hyperbolic measure $d\mu$. Fourth moment conjecture. As $t_g$ tends to infinity along a subsequence of real-valued newforms,

$$\int_{\Gamma_0(q)\backslash\mathbb{H}}|g(z)|^4\,d\mu(z)=\frac{3}{\operatorname{vol}(\Gamma_0(q)\backslash\mathbb{H})}+o_q(1).$$

This is the full-quotient $n=4$ case of the Gaussian moments conjecture and is expected to reflect Gaussian fourth moments. The source presents it as an open conjecture whose study is connected to spectral sums of $L$-functions.