The 2-sphere extension distinguishing number conjecture

Let $\mathbb{S}^2$ be the 2-sphere with its natural group action, and let $\operatorname{ext}_D(\mathbb{S}^2)$ denote its extension distinguishing number.

2-sphere extension conjecture.

$$\operatorname{ext}_D(\mathbb{S}^2)=9.$$

The conjecture is motivated by the faithful embedding of the 3-dimensional cube into $\mathbb{S}^2$ and the resulting lower bound; the exact value remains open.