The critical-diagonal conjecture for phase transition in the area-interaction model

Let $z\geq 0$ be the activity and let $\beta\geq 0$ be the inverse temperature of the area-interaction model. Phase transition means non-uniqueness of the area-interaction measures for these parameters. Critical-diagonal conjecture. There exists $0<\beta_c<\infty$ such that phase transition occurs if and only if

$$z=\beta>\beta_c.$$

The conjecture describes the still-unresolved phase diagram: existing results establish substantial bounds and an almost complete picture, but do not determine the full uniqueness/non-uniqueness region.