Isodiametric conjecture for Riesz capacity

Let $n\geq 2$, let $q(n)\leq n-2$, and let $K\subset\mathbb R^n$ be compact. Write $\operatorname{diam}(K)$ for its diameter and $\operatorname{Cap}_q(K)$ for its Riesz capacity. Isodiametric conjecture. A number $q(n)\leq n-2$ exists such that, whenever $q\in(q(n),n)$, the ratio

$$\frac{\operatorname{Cap}_q(K)}{\operatorname{diam}(K)}$$

is maximized by the ball among compact subsets of $\mathbb R^n$. The theorem immediately preceding the conjecture proves the endpoint cases $q=n-2$ and $q=n-1$ in related capacity inequalities, while the asserted wider range remains open; the source notes that Burchard, Choksi, and Hess-Childs rule out ball maximality for fixed positive $q$ in sufficiently large dimensions.