Bétermin's conjecture on minimizers of the Lennard-Jones lattice energy

Let $A$ denote the lattice-area parameter in the Lennard-Jones minimization problem

$$\min_{z\in\mathbb{H}}\bigl(\zeta(6,z)-b\zeta(3,z)\bigr),\qquad b\in\mathbb{R}.$$

The triangular lattice, rhombic lattices, a unique minimizer, and rectangular lattices are the candidate optimal shapes in the successive parameter regimes. Bétermin's conjecture. The solutions to the Lennard-Jones model are as follows: for $\frac{\pi}{(120)^{1/3}}<A<A_{BZ}\approx1.138$, the minimizer is triangular; for $A_{BZ}<A<A_1\approx1.143$, the minimizer is a rhombic lattice, continuously and monotonically covering the interval of angles $[76.43,90)$; for $A_1<A<A_2\approx1.268$, the minimizer is unique; and for $A>A_2$, the minimizer is a rectangular lattice. This conjecture gives the predicted phase diagram for optimal two-dimensional lattice shapes under the Lennard-Jones potential; the supplied text does not establish its resolution.