Gamow's conjecture for the Coulomb liquid drop model

Let $m_{\operatorname{ball}}$ be the critical mass below which balls minimize the liquid drop energy, let $m_\mathrm{ex}$ be the critical mass up to which minimizers exist, and let $m_\mathrm{crit}$ be the mass at which the energy of one ball equals twice that of a ball of half the mass. Gamow's conjecture. If $n=3$ and $\alpha=1$, then

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

Equivalently, the ball of mass $m$ is the unique minimizer up to translations for $mleq m_{\mathrm{crit}}$, and no minimizer exists for $m>m_{\mathrm{crit}}$. This conjecture is supported by known results on the liquid drop model and remains open in the Coulomb case; the paper notes that analogous equality is expected in arbitrary dimension and for every $0<\alpha<n$.