Supercritical Kasner-circle attractor conjecture

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, with stable set $S\subset\mathrm{K}^{\ocircle}$, attractor ${\cal A}_-$, and $\tau_-\to\infty$. Supercritical attractor conjecture. The stable set $S$ on $\mathrm{K}^{\ocircle}$ is the attractor ${\cal A}_-$. The conjecture concerns the global asymptotic behaviour of generic solutions: the source notes that local analysis shows attraction near $S$, while proving that all generic solutions have only points of $S$ as their $\omega$-limit remains open.