Global minimizer conjecture for the Morse potential lattice energy

Let $r_0>0$. For a Bravais lattice $L$ in dimension $3$, let $E_{\alpha,r_0}[L]$ denote the Morse lattice energy, let $\mathcal{L}_3$ be the class of Bravais lattices, and let $\mathcal{P}_3$ be the class of periodic structures. Let BCC, FCC, and HCP denote body-centered cubic, face-centered cubic, and hexagonal close-packed structures, respectively. Global minimizer conjecture. There exist parameters $\alpha_0$ and $\alpha_1$ such that:

1. If $\alpha<\alpha_0$, then the global minimizer of $E_{\alpha,r_0}$ in $\mathcal{P}_3$ is a BCC lattice.
2. If $\alpha>\alpha_1$, then the global minimizer of $E_{\alpha,r_0}$ in $\mathcal{L}_3$ is an FCC lattice, whereas the global minimizer in $\mathcal{P}_3$ is an HCP structure.
3. If $\alpha_0<\alpha<\alpha_1$, then the unique minimizer of $E_{\alpha,r_0}$ in $\mathcal{P}_3$ is an FCC lattice.

The conjecture predicts transitions among BCC, FCC, and HCP minimizers as the Morse parameter varies. The proposed transition is motivated by numerical computations and is heuristically justified in the paper for the BCC/FCC transition using a conjecture on lattice theta functions; the asserted global minimization statements remain open.