Moments of Riesz p-equilibrium measures are minimal for the ball

Let $n\geq 2$ and $n-2<p<n$, or let $n=1$ and $0\leq p<1$. Let $K\subset\mathbb{R}^n$ be compact and let $B\subset\mathbb{R}^n$ be a closed ball centered at the origin. Write $\operatorname{Cap}_p$ for the $p$-capacity, and let $\mu$ and $\nu$ be the $p$-equilibrium measures of $K$ and $B$, respectively. Assume that

$$\operatorname{Cap}_p(K)=\operatorname{Cap}_p(B).$$

Moments of Riesz $p$-equilibrium measures are minimal for the ball. For every $q\in(0,\infty)$,

$$\int_K |x|^q\,d\mu\geq\int_B |x|^q\,d\nu.$$

This extends the corresponding logarithmic and Newtonian moment inequalities and asserts that, among compact sets with fixed $p$-capacity, the centered ball minimizes every positive radial moment of the equilibrium measure. The supplied context does not state whether this conjecture has been proved or disproved.