Higher decorrelation conjecture for Hecke eigenforms

Let $\mathbb{H}=\{x+iy:x\in\mathbb{R},\ y>0\}$ be the upper half-plane, let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$, and let $H_k$ be a Hecke basis of holomorphic cusp forms on $\Gamma$. Let $\Omega$ be a fixed compact set of $\Gamma\backslash\mathbb{H}$ whose boundary $\partial\Omega$ has hyperbolic measure zero. Let $f,g\in H_k$ satisfy $\langle f,g\rangle=0$. Higher decorrelation conjecture. For every positive integer $a\in\mathbb{N}$,

$$\int_{\Omega}y^{ak}f(z)^a\overline{g(z)}^a\,\mathrm{d}\mu z=o(1)$$

as $k\to\infty$. This conjecture concerns higher mixed moments of two orthogonal Hecke eigenforms of equal weight and is intended to capture higher-order decorrelation, including joint sign-change information. Its resolution status is not specified in the supplied text.