  We confirm the list from \cite{MSS} of values $D$ for which the high-density
hard-core model on $\mathbb{Z}^2$ with exceptional distance $D$ has infinitely
many extremal Gibbs states. As a byproduct, we prove that for all $D>0$ there
exists an acute-angled triangle inscribes in $\mathbb{Z}^2$ with side-lengths
at least $D$ and area $\sqrt{3}/4\cdot D^2+O(D^{4/5})$.
