The super-Jacobian conjecture for affine superspaces

Let $\mathbb{F}$ be either $\mathbb{R}$ or an algebraically closed field of characteristic $0$, and consider the morphism $\phi$ of the affine superspace $\mathbb{A}^{m|n}$ over $\mathbb{F}$ introduced above. Write its even coordinate functions as $f_i$ and its odd coordinate functions as $q_k$, and let $\mathbf{J}$ denote the ideal appearing in the super-Jacobian condition. If

$$\mathsf{Jac}\left(\frac{\partial f_i}{\partial x_j}\right)\in \mathbb{F}^{\times}+\mathbf{J},$$

and

$$\mathsf{Jac}\left(\frac{\partial q_k}{\partial \xi_l}\right)\in \mathbb{F}^{\times}+\mathbf{J},$$

then $\phi$ is an automorphism. This is the proposed super version of the Jacobian conjecture; the paper verifies it under an additional preservation hypothesis on the set of maximal $\mathbb{Z}_2$-homogeneous ideals, but the general assertion remains unresolved.