Nonsplit quantum unique ergodicity conjecture for Hilbert–Maaß forms

Let $E$ be a real quadratic field, let $\Gamma$ be the relevant arithmetic subgroup, and let $(\phi_j)$ be an orthonormal basis of Hilbert Maaß cusp forms on $\mathrm{SL}_2(\mathcal{O}_E)\backslash(\mathbb{H}\times\mathbb{H})$, with spectral parameters $t_{1,j}$ and $t_{2,j}$. Restricting $\phi_j(z_1,z_2)$ to the diagonal gives the signed measure $d\mu_j(z)\coloneqq\phi_j(z,z)\,d\mu(z)$ on $\Gamma\backslash\mathbb{H}$. For $H\in C_b(\Gamma\backslash\mathbb{H})$, define

$$\mathcal{D}_j(H)\coloneqq \int_{\Gamma\backslash\mathbb{H}}H(z)\,d\mu_j(z)-\frac{3}{\pi}\mu_j(\Gamma\backslash\mathbb{H})\int_{\Gamma\backslash\mathbb{H}}H(z)\,d\mu(z).$$

Set

$$C(\operatorname{As}\phi_j)\coloneqq(3+|t_{1,j}+t_{2,j}|)^2(3+|t_{1,j}-t_{2,j}|)^2.$$

Nonsplit quantum unique ergodicity conjecture. For every fixed $H\in C_b(\Gamma\backslash\mathbb{H})$,

$$\lim_{C(\operatorname{As}\phi_j)\to\infty}\mathcal{D}_j(H)=0.$$

This is a nonsplit analogue of an off-diagonal quantum unique ergodicity statement: the diagonal restrictions of Hilbert Maaß cusp forms should dissipate as the Asai analytic conductor grows. The source says the conjecture is out of reach by current methods; a holomorphic base-change analogue is known, while the general case remains open.