Square-root threshold conjecture for two-type sparse random graphs

Let $c>1$ and $-c\leq\delta\leq c$ be constants, and let $obreak\kappa_{c+ obreak\delta,c- obreak\delta}$ be the two-type kernel defined in the source. Square-root threshold conjecture. If $obreak\delta<\sqrt c$, then the models $G_{1/n}(n, obreak\kappa_{c+ obreak\delta,c- obreak\delta})$ and $G(n,c/n)$ are essentially equivalent; if $obreak\delta>\sqrt c$, then they are not essentially equivalent. This would identify $obreak\sqrt c$ as the threshold for essential equivalence in this family; the source explicitly leaves the endpoint cases unresolved.