Square-lattice minimization conjecture for a difference of exponentials

Let $E_f(L)$ denote the lattice energy of a two-dimensional lattice $L$, and consider the potential $f(r)=e^{-\beta\pi r}-e^{-\alpha\pi r}$, where $\alpha$ and $\beta$ are parameters. Square-lattice minimization conjecture. The square lattice minimizes the lattice energy $E_f(L)$ when $|\beta-\alpha|$ is bigger than some small positive number. This is an open problem for a non-monotone potential, complementing questions about when the triangular lattice minimizes lattice energies beyond positive superpositions of Gaussians.