Regular simplex conjecture for Riesz capacity ratios

Let $n\geq 3$, let $p<q\leq -2$, and let $K\subset\mathbb R^n$ be compact with more than one point. The quantities $\operatorname{Cap}_p(K)$ and $\operatorname{Cap}_q(K)$ are the corresponding Riesz capacities. Regular simplex conjecture. Among such sets $K$, the regular $(n+1)$-point set maximizes

$$\frac{\operatorname{Cap}_q(K)}{\operatorname{Cap}_p(K)}.$$

The claim is known for $n=1$ and $n=2$, while the proposition handling the diameter edge case gives related results for $q\leq-2$; the stated ratio conjecture for $n\geq3$ remains open.