The L4-norm problem for Hecke–Maaß eigenforms

Let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$, let $\mathbb{H}$ be the upper half-plane, and let $g\in\mathcal{B}_0(\Gamma)$ be a Hecke–Maaß eigenform normalised by $\langle g,g\rangle=1$. The $L^4$-norm problem. As $t_g\to\infty$,

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

This is the second nontrivial case of the Gaussian moments conjecture and is closely related to quantum unique ergodicity. The source states that an unconditional proof appears quite difficult; the conjecture therefore remains open.