Periodic ABC arrangement conjecture for the ternary long-range Ising model

Consider a one-dimensional periodic cell containing $n$ balls of each type $A$, $B$, and $C$, with positive radii parameters $\omega_i$ satisfying $\sum_i\omega_i=1$, and let $[f_{ij}]$ be an admissible interaction strength matrix. Denote by $U$ the total pairwise electrostatic potential energy. Periodic ABC arrangement conjecture. For every admissible matrix $[f_{ij}]$ and every positive collection $\{\omega_i\}$ with $\sum_i\omega_i=1$, the periodic arrangement

$$ABC\,\cdots\,ABC$$

that is, $i_k\equiv k\pmod 3$, minimizes $U$. This is a proposed ground-state characterization for the ternary one-dimensional long-range Ising model. The supplied text does not give a proof or resolution, and the precise meaning of admissibility is deferred to the paper's earlier conditions.