Minimality conjecture for the BPD energy in the plane

Let $N\in\mathbb N\backslash\{0,1\}$ and let $X_N\in(\mathbb R^2)^N$ be a configuration of $N$ points. For the potential $V_{\textnormal{BPD}}$ and its energy $\textnormal{E}_{\textnormal{BPD}}$, let $Z_N$ denote the specified square-lattice configuration and let $\mathcal{E}_{\|\cdot\|_\infty}(N)$ be the minimum energy for the corresponding norm energy on $\mathbb Z^2$.

Minimality for $\textnormal{E}_{\textnormal{BPD}}$ in general. For all $N\in\mathbb N\backslash\{0,1\}$, we have

$$\min_{X_N\in(\mathbb R^2)^N}\textnormal{E}_{\textnormal{BPD}}(X_N)=\mathcal{E}_{\|\cdot\|_\infty}(N)=-\lfloor4N-\sqrt{28N-12}\rfloor,$$

achieved in particular for $X_N=Z_N$.

This conjecture asserts crystallization of finite minimizers for the BPD potential: the minimum over arbitrary planar configurations equals the square-lattice minimum, with a square-lattice minimizer. The corresponding minimality result is known when configurations are constrained to $\mathbb Z^2$, while the unrestricted finite-$N$ crystallization statement remains open.