Exact TISGM count conjecture for the ferromagnetic hard-core and soft-core Widom–Rowlinson model

Let $k\geq 4$ and consider the ferromagnetic model with critical values $\theta'_{\rm c}(k)$, $\lambda'_{\rm cr}(k)$, $\theta_{\rm c}(k)$, and $\lambda_{\rm cr}(k)$ as defined in the preceding theorem. A TISGM is a translation-invariant splitting Gibbs measure. Exact TISGM count conjecture. In part 2) of Theorem, the number of TISGMs is exactly one, and in part 3), the number of TISGMs is exactly three. This conjecture sharpens the preceding existence results by replacing “at least” with exact counts in the corresponding parameter regimes; the supplied text gives no resolution.