The circle extension distinguishing number conjecture

Let $\mathbb{S}^1$ be the circle with its natural group action, and let $\operatorname{ext}_D(\mathbb{S}^1)$ denote its extension distinguishing number.

Circle extension conjecture.

$$\operatorname{ext}_D(\mathbb{S}^1)=6.$$

The paper proves the lower bound $\operatorname{ext}_D(\mathbb{S}^1)\geq 6$ and conjectures that this lower bound is sharp; the matching upper bound is left open.