Conjecture on the image of the motivic birational invariant for projective four-space

Let $c$ be the motivic birational invariant considered in the paper, and let $S$ and $S'$ be D-equivalent K-nef surfaces. The corresponding elements are the differences $[\mathbb P^1]([S]-[S'])$. Image-generation conjecture. The image $c({\operatorname{Bir}}(\mathbb P^4_\mathbb C))$ is generated by elements of the form

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

This proposes a description of the currently unknown image of $c$ in the simplest nontrivial case. Known examples arise from D-equivalent K3 surfaces and D-equivalent elliptic surfaces of Kodaira dimension $1$; the conjecture would follow from a suitable lifting of the theory of Hodge atoms to derived categories, a statement currently known only in low dimension.