Kasner–Bianchi II attractor conjecture

Let $\mathcal{A}$ denote the attractor describing the generic behaviour near the initial singularity, let $\mathcal{K}$ be the Bianchi type I Kasner subset, and let $\mathcal{B}_{\mathrm{II}}$ be the Bianchi type II subset. Let $\mathrm{K}^{\circ}$ and the subsets $\mathcal{T}_{R_1}$, $\mathcal{T}_{R_3}$, $\mathcal{T}_{R_1R_3}$, $\mathcal{T}_{N_-}$, and $\mathcal{T}_{R_1N_-}$ denote the corresponding invariant pieces. Kasner–Bianchi II attractor conjecture. The attractor is the union of the Bianchi type I Kasner and Bianchi type II subsets, namely

$$\mathcal{A}=\mathcal{K}\cup\mathcal{B}_{\mathrm{II}}=\mathrm{K}^{\circ}\cup\mathcal{T}_{R_1}\cup\mathcal{T}_{R_3}\cup\mathcal{T}_{R_1R_3}\cup\mathcal{T}_{N_-}\cup\mathcal{T}_{R_1N_-}.$$

The claim identifies the full attractor governing generic behaviour near the initial singularity with these Kasner and single-transition pieces; the source gives no evidence that it has been resolved.