Higher-dimensional Gamow conjecture for Riesz interactions

Let $n\geq 2$ and $0<\alpha<n$, and let $m_{\operatorname{ball}}$, $m_\mathrm{ex}$, and $m_\mathrm{crit}$ denote respectively the critical mass below which balls minimize the liquid drop energy, the critical mass up to which minimizers exist, and the mass at which the energy of one ball equals twice that of a ball of half the mass. Gamow's conjecture.

$$m_{\operatorname{ball}}=m_\mathrm{ex}=m_\mathrm{crit}.$$

The paper presents this as the expected extension of the Coulomb conjecture to arbitrary dimension and Riesz exponent, but the supplied text gives no resolution.