Conjecture on the motivic birational invariant for projective three-space over arbitrary fields

Let $\mathbb k$ be a field, and let $C$ and $C'$ be smooth projective curves over $\mathbb k$ that are D-equivalent and whose geometric irreducible components have genus $1$. The associated differences are $\mathbb P^1\cdot([C]-[C'])$. Image-generation conjecture. For any field $\mathbb k$, the image $c({\operatorname{Bir}}(\mathbb P^3_\mathbb k))$ is generated by differences of the form

$$\mathbb P^1\cdot([C]-[C']).$$

This is a conjectural description over nonclosed fields, where the invariant is known to vanish over algebraically closed fields of characteristic zero. The proposed generators reflect the currently constructed nontrivial examples; only curves whose geometric irreducible components have genus at most $1$ can contribute, and genus-zero curves yield no nontrivial D-equivalence.