The compatibility conjecture for the motivic group structure on projective-line endomorphisms

Let $\mathcal{J}$ be the Jouanolou device of $\mathbb{P}^1$, and let $[\mathcal{J},\mathbb{P}^1]^\mathrm{N}$ and $[\mathbb{P}^1,\mathbb{P}^1]^{\mathbb{A}^1}$ denote the corresponding naive and motivic homotopy classes. The bijection

$$\xi \colon [\mathcal{J},\mathbb{P}^1]^\mathrm{N} \to [\mathbb{P}^1,\mathbb{P}^1]^{\mathbb{A}^1}$$

identifies the proposed group operation $\oplus$ on the source with the conventional motivic group operation $\oplus^{\mathbb{A}^1}$ on the target. The compatibility conjecture. The bijection $\xi$ is a group isomorphism and equals $\phi$. Establishing this would identify the explicit group structure on naive maps from the Jouanolou device with the standard motivic group structure on endomorphisms of the projective line; the paper proves that $\oplus$ makes the source an abelian group and constructs the isomorphism $\phi$, but compatibility with $\xi$ remains unproved.