Loop-constrained regular polygon maximization conjecture for quantum dot families

Let $Y$ be a finite set of points on a loop $\Gamma$ of fixed length in $\mathbb{R}^{\nu}$, where $\nu=2,3$, with the points equidistant in the arc-length variable, and suppose that the balls $B_{\rho}(y_i)$ do not overlap. Write $H_{V,Y}$ for the associated Schrödinger operator and let $\epsilon_1(H_{V,Y})=\inf\sigma(H_{V,Y})$. Loop maximization conjecture. The quantity $\epsilon_1(H_{V,Y})$ is maximized, uniquely up to Euclidean transformations, when $Y$ is the vertex set of a planar regular polygon with $\#Y$ vertices. This extends the established circular setting to arbitrary loops and asks how geometry affects the ground-state energy in broader quantum dot families.