Uniqueness conjecture for relative motivic spheres in dimension 2

Let $S$ be any Noetherian scheme with characteristic zero residue fields, and let $f:X\to S$ be a smooth scheme of relative dimension $2$.

Uniqueness conjecture for relative motivic spheres. If $X$ is $\mathbb{A}^1$-homotopic to $\mathbb{A}^2_S\backslash\{0\}$, then $X$ is isomorphic to $\mathbb{A}^2_S\backslash\{0\}$ as an $S$-scheme.

This conjecture would assert that there are no exotic motivic spheres in relative dimension $2$. It is proposed as a possible formulation in anticipation of a gluing lemma for motivic spheres; the source does not indicate whether it is known or resolved.