Stable-set conjecture for supercritical Cantor-set heteroclinic chains

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, let $C$ be the Cantor set associated with infinite Bianchi type II heteroclinic chains, and let $\tau_-\to\infty$. Supercritical-chain stable-set conjecture. Each infinite heteroclinic chain associated with $C$ has a stable invariant set of co-dimension one. The conjecture extends the period-two case to the infinite chains associated with the Cantor set and is intended to describe their asymptotic continuous dynamics; no proof is supplied here.