Kazhdan–Yom Din's asymptotic Schur orthogonality conjecture

Let $G$ be a locally compact group with Lie algebra ${\mathfrak g}$, equipped with a norm and the induced operator norm on $\operatorname{End}({\mathfrak g})$. Define the radius function

$$\mathbf{r}(g)=\log\left(\max\{\|\operatorname{Ad}(g)\|,\|\operatorname{Ad}(g^{-1})\|\}\right),$$

and let $G_{<r}$ be the set of elements $g\in G$ such that $\mathbf{r}(g)<r$. Let $V$ be a tempered irreducible unitary $G$-representation.

Asymptotic Schur orthogonality relations. There exist $\mathbf{d}(V)\in{\mathbb Z}_{\ge 0}$ and $\mathbf{f}(V)\in{\mathbb R}_{>0}$ such that, for all $v_1,v_2,v_3,v_4\in V$,

$$\lim_{r\to+\infty}\frac{\int_{G_{<r}}\langle gv_1,v_2\rangle\overline{\langle gv_3,v_4\rangle}\,dg}{r^{\mathbf{d}(V)}}=\frac{1}{\mathbf{f}(V)}\langle v_1,v_3\rangle\overline{\langle v_2,v_4\rangle}.$$

This proposes a polynomially normalized asymptotic orthogonality law for matrix coefficients of tempered irreducible unitary representations. The supplied text gives the formulation but no evidence resolving it.