Real Jacobian pairs with components of low degrees

We prove that every polynomial map $(f,g):{\mathbb{R}}^2\to {\mathbb{R}}^2$ with nowhere vanishing Jacobian such that $\mathrm{d}\mathrm{e}\mathrm{g}f\le 5$, $$\mathrm{d}\mathrm{e}\mathrm{g}$,g \leq 6$ is injective.