Single-transition attractor conjecture for Bianchi type VI$_{-1/9}$

Let $\mathcal{A}$ denote the attractor describing generic behaviour near the initial singularity, and let $\mathrm{K}^{\circ}$, $\mathcal{T}_{R_1}$, $\mathcal{T}_{R_3}$, and $\mathcal{T}_{N_-}$ denote the Kasner subset and the indicated transition subsets. Single-transition attractor conjecture. The attractor is

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

This is the stronger conjecture motivated by numerical evidence that multiple transitions become increasingly rare, so that only single transitions contribute asymptotically; the source also notes supporting analytical and cosmological-billiard arguments, but gives no resolution.