Stable-set conjecture for period-two heteroclinic cycles

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, with time parameter satisfying $\tau_-\to\infty$. Period-two stable-set conjecture. Each heteroclinic cycle with period $2$ has a stable invariant set of co-dimension one. This is presented as the first step in a proposed hierarchy of conjectures for connecting the discrete heteroclinic dynamics with the asymptotic continuous dynamics; no proof is supplied here.