Uniqueness of the Gibbs measure at activity squared four

Let $D$ denote the activity parameter of the high-density hard-core model on $\mathbb{Z}^3$, and let a Gibbs measure be an infinite-volume equilibrium measure for this model. Uniqueness conjecture. For $D^2=4$ there is a unique Gibbs measure. The conjecture concerns the phase-transition behaviour at the threshold where the Peierls bound fails and many periodic configurations occur; whether the Gibbs measure is unique at this parameter remains open.