Joint mass distribution 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 $J\in\mathbb{Z}_{>1}$, let $f_j\in H_{k_j}$ satisfy

$$\langle f_j,f_j\rangle=\operatorname{vol}(\Gamma\backslash\mathbb{H})\qquad(1\leq j\leq J),$$

and assume $\langle f_i,f_j\rangle=0$ whenever $k_i=k_j$ and $i\ne j$. For $a=(a_j)\in\mathbb{Z}_{\geq1}^J$, joint mass distribution conjecture.

$$\frac{1}{\operatorname{vol}(\Omega)}\int_{\Omega}y^{\sum_{j=1}^J a_jk_j}\prod_{j=1}^J|f_j(z)|^{2a_j}\,\mathrm{d}\mu z=\prod_{j=1}^J(a_j)!+o(1)$$

as $\min k_j\to\infty$. This asserts that orthogonal Hecke eigenforms are statistically independent in the large-weight limit, extending the single-form locally Gaussian value distribution conjecture. The supplied text does not state whether the conjecture has been resolved.