The one-dimensional point-interaction ground-state optimization conjecture

Let $Y$ and $\tilde{Y}$ be $N$-point configurations in the described one-dimensional setting, with $\alpha\in(0,\infty)$ denoting the interaction strength. Two configurations are congruent if they are related by an isometry of the underlying circle.

Ground-state optimization conjecture. The inequality

$$\lambda_1(\alpha,Y)<\lambda_1(\alpha,\tilde{Y})$$

holds for any $\alpha\in(0,\infty)$ unless the $N$-point configurations $Y$ and $\tilde{Y}$ are congruent.

The preceding weak- and strong-coupling results motivate this conjecture: noncongruent configurations should be strictly ordered by their lowest eigenvalue for every positive interaction strength. Its status is not resolved in the supplied text.