The beta-catastrophe conjecture for emergence of a giant wandering strongly connected component

Let $G_{\beta,n}$ be the support digraph for a discrete family, let $V_n$ be its vertex set, and let $S_n$ be the set of solution states. Write

$$\Phi_n(\beta):=\frac{1}{|V_n|}\left|\operatorname{SCC}_{\max}\bigl(G_{\beta,n}\restriction(V_n\setminus S_n)\bigr)\right|.$$

A phase transition at $\beta_c\in(0,1)$ means that $\lim_{n\to\infty}\Phi_n(\beta)=0$ for $\beta<\beta_c$ and $\liminf_{n\to\infty}\Phi_n(\beta)>0$ for $\beta>\beta_c$. The $\beta$-catastrophe conjecture. For certain hard discrete families, there exists $\beta_c\in(0,1)$ such that $\Phi_n(\beta)$ undergoes a phase transition at $\beta_c$. For $\beta>\beta_c$, typical trajectories exhibit long transients supported by the giant strongly connected component of $G_{\beta,n}\restriction(V_n\setminus S_n)$ before reaching $S_n$, causing rapid growth or divergence of solution times as $\beta\uparrow1$. The claim connects a graph phase transition in the wandering region to critical slowing down or failure of the associated discrete dynamics; the source does not state whether it has been proved or refuted.