Single HCP equivalence class conjecture for $D^2=2\ell^2$ divisible by 3

Suppose $D^2=2\ell^2$ with $\ell\in{\mathbb N}$ and $\ell\equiv0\pmod 3$. Let a PC be a periodic configuration and an EPGM an extremal periodic Gibbs measure. Two periodic configurations are understood to belong to the same equivalence class under the equivalence relation used for periodic configurations in the paper. Write $\mathscr E(D,u)$ for the set of EPGMs at fugacity $u$.

Single-class HCP conjecture. All EPGMs are generated by PCs from a single equivalence class. This class has cardinality $16\ell^3$ and consists of $8$ $D$-HCP configurations and their ${\mathbb Z}^3$-shifts. Consequently,

$$\sharp\mathscr E(\sqrt 2\ell,u)=16\ell^3.$$

The conjecture refines the expected selection of HCP configurations by specifying the unique generating equivalence class and the resulting number of EPGMs. The paper states that the general proof is not known, although the analogous case $D^2=5$ was verified.