Global minimality conjecture for FCC and BCC lattices under Lennard–Jones-type energies

Let $f$ be the Lennard–Jones-type interaction defined by

$$f(r)=\frac{a_2}{r^{x_2}}-\frac{a_1}{r^{x_1}},$$

where $(a_1,a_2)\in(0,+\infty)^2$ and $0<x_1<x_2$. Let $E_f$ denote the interaction energy among Bravais lattices, and let FCC and BCC refer respectively to the face-centered cubic and body-centered cubic lattices. Global minimality conjecture. If $3/2<x_1<x_2$, then the unique minimizer of $E_f$ among all Bravais lattices is an FCC lattice. If $0<x_1<x_2<3/2$, then the unique minimizer of $E_f$ among all Bravais lattices is a BCC lattice. This conjecture concerns global minimization without a volume constraint; the preceding local results establish FCC and BCC local optimality in the appropriate high- and low-density regimes, while the global minimization problem remains open in the stated generality.