Equidistribution of orthogonal grids on integer spheres

For $d\geq 3$, let $\mathbb{S}^{d-1}(D)=\{v\in\mathbb{Z}^d:\|v\|^2=D\}$, and for each primitive vector $v\in\mathbb{S}^{d-1}(D)$ let $[\Delta_v]\in\mathcal{Y}_{d-1}$ be the rotation class of the grid formed by the orthogonal lattice $\Lambda_v=\mathbb{Z}^d\cap v^\perp$ together with the orthogonal projection of a vector $w\in\mathbb{Z}^d$ satisfying $(w,v)=1$. Let $\widetilde{\nu}_D$ be the normalized counting measure on

$$\left\{\left(\frac{v}{\|v\|},[\Delta_v]\right):v\in\mathbb{S}^{d-1}(D)\right\}\subset\mathbb{S}^{d-1}\times\mathcal{Y}_{d-1},$$

where

$$\mathcal{Y}_{d-1}=\operatorname{SO}_{d-1}(\mathbb{R})\setminus\operatorname{ASL}_{d-1}(\mathbb{R})/\operatorname{ASL}_{d-1}(\mathbb{Z}).$$

Equidistribution conjecture for orthogonal grids. The convergence

$$\widetilde{\nu}_D\stackrel{\mathrm{weak}^{*}}{\longrightarrow}m_{\mathbb{S}^{d-1}}\otimes m_{\mathcal{Y}_{d-1}}$$

holds for the subset $A=\mathbb{N}$ if $d>4$, for $A=\mathbb{N}\setminus(8\mathbb{N})$ if $d=4$, and for

$$A=\{D\geq 1\mid D\text{ is not congruent to }0,4,7\text{ modulo }8\}$$

if $d=3$, as $D\to\infty$ with $D\in A$. Here $m_{\mathbb{S}^{d-1}}$ and $m_{\mathcal{Y}_{d-1}}$ are the natural uniform probability measures on the sphere and the grid-moduli space. This generalizes Linnik's problem on spheres by incorporating the shapes and marked points of the orthogonal grids; the stated parser status is unknown, so the resolution status is left open.