Infinite Kinser class self-duality conjecture

Let $\mathcal{K}_\infty$ denote the class obtained from the hierarchy of Kinser classes, and let $\mathcal{K}_\infty^*$ denote its dual class. Infinite Kinser class self-duality conjecture.

$$\mathcal{K}_\infty^*=\mathcal{K}_\infty.$$

The source states that this conjecture would follow from the preceding dual-containment conjecture. It therefore remains open unless that preceding conjecture is proved.