Higher-cardinality universal optimality conjecture for rectangular flat tori

For $\beta>0$, let

$$\Phi_\beta:=\begin{bmatrix}1&1/2\\0&\beta/2\end{bmatrix}\mathbb{Z}^2,$$

and, for $m\geq 2$, define

$$\omega^*_{m,\beta}:=\Phi_\beta\cap([0,1]\times[0,m\beta]),\qquad L^m_\beta:=\begin{bmatrix}1&0\\0&m\beta\end{bmatrix}\mathbb{Z}^2.$$

A configuration is $L^m_\beta$-universally optimal if it minimizes every admissible periodic energy among $2m$-point configurations periodic with respect to $L^m_\beta$. Higher-cardinality universal optimality conjecture. For all $m\geq 2$, $\omega^*_{m,\beta}$ is $L^m_\beta$-universally optimal if and only if

$$\beta\geq 1/\sqrt{3}.$$

The conjecture is motivated by the universal optimality conjecture for $A_2$ and extends the paper's proved four-point case $m=2$; the cases $m\geq 2$ are not established here.