Orbit-independence conjecture for elementary abelian subalgebra varieties

Let $G$ be a reductive algebraic group with a Chevalley $\mathbb{Z}$-form $G_{\mathbb{Z}}$, and set

$$G_p=G_{\mathbb{Z}}\times_{\mathbb{Z}}\operatorname{Spec}\overline{\mathbb{F}_p}.$$

Let $(\star)$ be the condition on primes used in the paper, and let $d$ be defined by the finite-field extensions in question. For a map $\varphi:G\to G'$ of reductive groups defined over $\mathbb{Z}$, write $d\varphi$ for the induced map on orbits. Orbit-independence conjecture. (1) For every reductive algebraic group $G$ and every pair of primes $p,p'$ satisfying $(\star)$, there is a natural dimension-preserving bijection $f_{p,p'}$ between the $G_p$-orbits of $\mathbb{E}(r,\operatorname{Lie}(G_p))$ defined over $\mathbb{F}_{p^d}$ and the $G_{p'}$-orbits of $\mathbb{E}(r,\operatorname{Lie}(G_{p'}))$ defined over $\mathbb{F}_{p'^d}$. Naturality means that for primes $p,p',p”$ satisfying $(\star)$,

$$f_{p,p”}=f_{p',p”}\circ f_{p,p'},$$

and, for every such $\varphi:G\to G'$,

$$d\varphi\circ f_{p,p'}=f'_{p,p'}\circ d\varphi.$$

(2) If $\mathcal{O}_p$ is a $G_p$-orbit of $\mathbb{E}(r,\operatorname{Lie}(G_p))$ of dimension $e$, defined over $\mathbb{F}_{p^d}$, then for all primes $p'$ satisfying $(\star)$, the function sending $p'$ to $\#f_{p,p'}(\mathcal{O}_p)(\mathbb{F}_{p'})$ is a polynomial in $p'$ of degree $ed$. The conjecture would make orbit dimensions computable from the degree of the polynomial counting rational points, but the source does not provide a resolution.