Real topological Jacobian conjecture

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a real polynomial map with Jacobian determinant $f'(x)>0$ for every $x\in\mathbb{R}^n$, and let $S_f$ denote its non-properness set. Real topological Jacobian conjecture. If

$$\operatorname{codim} S_f\geq 2,$$

then $f$ is a bijection and consequently $S_f=\varnothing$. The provided text presents this as a proposed topological version of the real Jacobian conjecture and gives no resolution status.