The divergence criterion for polynomial inverse maps

Let $a(z)\in {\mathbb C}_2[[z_1,z_2,\ldots,z_n]]$ satisfy

$$\triangledown a(z)=0.$$

Define the formal differential operator $A=a(z)\frac{\partial}{\partial z}$ and the formal maps

$$F(z)=\exp(A)z,\qquad G(z)=\exp(-A)z.$$

Divergence criterion. The map $F(z)$ is polynomial if and only if $G(z)$ is polynomial. The source presents this as an algebraic formulation equivalent to the Jacobian conjecture, so it remains open.