The equality conjecture for the explicit subgroup of naive projective-line maps

Let $\mathcal{J}$ be the Jouanolou device of $\mathbb{P}^1$, and let $\mathbf{G}_0\subseteq\mathbf{G}$ be the subgroups introduced from the explicit generators, with $[\mathcal{J},\mathbb{A}^2\setminus\{0\}]^\mathrm{N}$ and $[\mathcal{J},\mathbb{P}^1]^\mathrm{N}$ denoting naive homotopy classes. Equality conjecture. The inclusions

$$\mathbf{G}_0 \subseteq [\mathcal{J},\mathbb{A}^2\setminus \{0\}]^\mathrm{N}$$

and

$$\mathbf{G} \subseteq [\mathcal{J},\mathbb{P}^1]^\mathrm{N}$$

are equalities. This conjecture would show that the explicitly generated groups exhaust the relevant naive homotopy classes; the paper proves the corresponding assertion over finite fields for the degree-zero part through a computation of $K_1^{MW}(\mathbb{F}_q)$, while the general equality remains open.