The plane extension distinguishing number conjecture

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

Plane extension conjecture.

$$\operatorname{ext}_D(\mathbb{R}^2)=7.$$

The value is motivated by lower bounds from faithful graph embeddings and by the relation between extension distinguishing numbers of spheres and Euclidean spaces; determining the exact value remains open.