Bétermin's conjecture on phase transitions for one-well potentials

Let $L$ be a two-dimensional lattice of area $A$, and let $E_f(L)$ denote its lattice energy for a potential $f$. A potential is completely monotone if it satisfies $(-1)^j f^{(j)}(x)>0$ for all $j=0,1,2,\ldots$, and it is one-well if there exists $a>0$ such that $f$ is nonincreasing on $(0,a)$ and nondecreasing on $(a,+\infty)$. A Lennard--Jones type potential has the form $f(r)=a_1r^{-t_1}-a_2r^{-t_2}$, where $t_1>t_2>0$ and $a_1,a_2>0$. Bétermin's conjecture. The behavior of the minimizers of $E_f(L)$ with respect to the lattice area $A$ is qualitatively the same for all Lennard--Jones type potentials, namely, the minimizers undergo hexagonal--rhombic--square--rectangular lattice phase transitions. More generally, the same result should hold for every potential $f=f_1-f_2$ such that $f_1$ and $f_2$ are completely monotone and $f$ is one-well. This conjecture extends the known triangular-lattice minimization result for completely monotone potentials to non-monotone one-well potentials and predicts a universal sequence of lattice phase transitions; the general assertion remains open.