Partial-symmetry symmetrization conjecture for maximizers

Let $({\mathcal L},{\mathbf e})$ satisfy the appropriate nondegeneracy and admissibility hypotheses, and suppose that $\mathcal L$ satisfies the partial symmetry hypothesis. A maximizer is a tuple $\mathbf E$ of sets, and a translate is obtained by translating its component sets by a common vector. Partial-symmetry symmetrization conjecture. For generic $({\mathcal L},{\mathbf e})$ satisfying the partial symmetry hypothesis, any maximizer $\mathbf E$ is a translate of a maximizer satisfying

$$\mathbf E={\mathbf E}^{\dagger}.$$

The conjecture would reduce the study of generic maximizers under partial symmetry to symmetrized maximizers; it is not proved in the paper.