Strong real Jacobian conjecture

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a real polynomial map, and let $f'(x)=\det\operatorname{Jac}_x f$. Strong real Jacobian conjecture. If

$$f'(x)>0$$

for every $x\in\mathbb{R}^n$, then $f$ is invertible. This conjecture is refuted: Pinchuk constructed a non-injective polynomial map in dimension two whose Jacobian determinant is everywhere positive.