Large-activity monochromaticity conjecture for the non-integrable Widom–Rowlinson model

Consider the Widom–Rowlinson model with $q$ types, activity $z$, and radius distributions $Q_i$ in the non-integrable case, meaning that there exists $1\leq i\leq q$ such that

$$\int_{\mathbb{R}^+} r^d Q_i(dr)=\infty.$$

Large-activity monochromaticity conjecture. For activities $z$ large enough, every Widom–Rowlinson measure is monochromatic. The paper proves the existence of polychromatic phases for sufficiently small activity in the completely non-integrable setting, whereas this asserted large-activity monochromaticity remains open.