Uniqueness of symmetrized maximizers under partial symmetry

Let $({\mathcal L}^0,\mathbf e)$ be nondegenerate and strictly admissible. For generic admissible $({\mathcal L},\mathbf e)$ satisfying the partial symmetry hypothesis, with $\mathcal L$ sufficiently close to $\mathcal L^0$, consider maximizers of $\Lambda_{\mathcal L}$ satisfying

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

A measure-preserving dilation of $\mathbb R^2$ is a dilation preserving Lebesgue measure. Uniqueness conjecture for symmetrized maximizers. Such maximizers are unique up to measure-preserving dilations of $\mathbb R^2$. This is a local generic uniqueness assertion beyond the perturbative existence and regularity results proved in the paper, and remains open.