Zhou's orthogonality relation for Maass forms on GL(n)

Fix $n\geq 2$. Let $\{\phi_j\}_{j=1,2,\ldots}$ be an orthogonal basis of Maass cusp forms for $\operatorname{SL}(n,\mathbb Z)$, with Langlands parameters $\alpha^{(j)}$ and normalized Fourier coefficients $A_j(M)$, and let $\mathcal L_j$ be the residue at $s=1$ of $L(s,\phi_j\times\overline{\phi_j})$. For $M=(m_1,\ldots,m_{n-1})$ and $M'=(m'_1,\ldots,m'_{n-1})$ in $\mathbb Z^{n-1}_{+}$, and a good test function $h_T$, define $\delta_{M,M'}=1$ if $M=M'$ and $0$ otherwise. Zhou's orthogonality conjecture. For some constant $0<\theta<1$,

$$\sum_{j=1}^\infty A_j(M)\overline{A_j(M')}\frac{h_T\left(\alpha^{(j)}\right)}{\mathcal L_j}=\delta_{M,M'}\sum_{j=1}^\infty\frac{h_T\left(\alpha^{(j)}\right)}{\mathcal L_j}+\mathcal O_{M.M'}\left(\sum_{j=1}^\infty\frac{h_T\left(\alpha^{(j)}\right)}{\mathcal L_j}\right)^\theta.$$

This error-term formulation implies the corresponding normalized orthogonality relation as $T\to\infty$. The relation describes asymptotic orthogonality of Fourier coefficients in the spectral family of Maass cusp forms for $\operatorname{SL}(n,\mathbb Z)$. The source states that the relation was proved for $n=3$ by Goldfeld and Kontorovich, so the conjecture as stated for general $n\geq2$ is resolved according to the supplied status evidence.