The 3-dimensional Euclidean extension distinguishing number conjecture

Let $\mathbb{R}^3$ be three-dimensional Euclidean space with its natural group action, and let $\operatorname{ext}_D(\mathbb{R}^3)$ denote its extension distinguishing number.

3-dimensional Euclidean extension conjecture.

$$\operatorname{ext}_D(\mathbb{R}^3)=10.$$

The conjecture is posed as part of the paper's program of determining extension distinguishing numbers for Euclidean spaces and spheres; no general upper bound establishing the equality is given here.