Hua–Huang–Li's statistical independence conjecture for powers of Hecke–Maass forms

Let $\{f_i\}_{i=1}^I$ be an orthogonal family of Hecke–Maass cusp forms, with $\langle f_i,f_{i'}\rangle=0$ for $i\ne i'$, and let $C_n$ denote the $n$th moment of a standard real Gaussian. Hua–Huang–Li's conjecture. For positive integers $a_i$, the powers $\{f_i^{a_i}\}_{i=1}^I$ are statistically independent: for every $\psi\in C_c^\infty(\mathbb{X})$,

$$\int_{\mathbb{X}}\psi(z)\prod_{i=1}^I f_i^{a_i}(z)\frac{\mathrm{d}x\,\mathrm{d}y}{y^2}\sim\prod_{i=1}^I C_{a_i}\int_{\mathbb{X}}\psi(z)\frac{\mathrm{d}x\,\mathrm{d}y}{y^2}$$

as $\min\{t_{f_1},\ldots,t_{f_I}\}$ tends to infinity. This extends the Gaussian moments prediction to joint moments of an orthogonal family; the source attributes it to Hua, Huang, and Li.