The scaling-group conjecture for unimodular solvable algebraic groups over $\mathbb{Q}_n$

Let $G$ be a unimodular solvable algebraic group over $\mathbb{Q}_n$, and let $p_1,\ldots,p_k$ be the primes appearing in the prime decomposition of $n$. Its scaling group is the subgroup of positive real numbers arising from measure-scaling quasi-isometries of $G$.

Scaling-group conjecture. The scaling group of $G$ is a subgroup, possibly trivial, of

$$\{p_1^{n_1}p_2^{n_2}\ldots p_k^{n^k}\mid n_1,\ldots,n_k\in\mathbb{Z}\}.$$

This proposed description concerns the possible scaling factors for unimodular solvable algebraic groups over non-Archimedean local fields. The surrounding discussion indicates that the claim is motivated by algebraic constraints on automorphism scaling groups, but gives no resolution.