Subcritical and critical Kasner-circle attractor conjecture

Consider Bianchi type VIII and IX models in the subcritical and critical cases $v\in(0,1/2]$, and let ${\cal A}_-$ denote the attractor as $\tau_-\to\infty$. Define the invariant subsets $\mathrm{II}_1$, $\mathrm{II}_2$, and $\mathrm{II}_3$, together with the Kasner circle $\mathrm{K}^{\ocircle}$. Subcritical and critical attractor conjecture. The attractor ${\cal A}_-$ consists of

$$\mathrm{II}_1\cup\mathrm{II}_2\cup\mathrm{II}_3\cup\mathrm{K}^{\ocircle}.$$

This conjecture proposes the global asymptotic attractor in the subcritical and critical regimes; the source supplies no proof or resolution.