The triangular lattice minimizer conjecture for logarithmic lattice energies

Let $s>2$. For a two-dimensional lattice $L$ of unit density, define

$$\mathcal{F}_s(L):= -\sideset{}{'} \sum_{p\in L} \frac{s\log |p|+1}{|p|^s}.$$

Triangular lattice minimizer conjecture. For every $s>2$, the triangular lattice $\mathsf{A}_2$ is the unique minimizer of $\mathcal{F}_s$ among all two-dimensional lattices of unit density.

This conjecture would imply the paper's optimality results for the triangular lattice for every pair of Lennard–Jones exponents $2<\beta<\alpha$, rather than only for fixed exponents. The source presents it as expected to hold for all $s>2$ and does not report a proof or disproof.