Jelonek's conjecture on polynomial mappings with non-properness set of codimension at least two

Let $F=(f_1,\ldots,f_n):\mathbb R^n\to\mathbb R^n$ be a polynomial mapping. Write $J(F)$ for its Jacobian determinant and $S_F$ for its non-properness set, and let $\operatorname{codim} S_F$ denote its codimension. Jelonek's conjecture. If $J(F)$ is nowhere vanishing and

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

then $F$ is a bijective mapping. Jelonek proved the claim under the stronger assumption $\operatorname{codim} S_F\geq3$ and in dimension $n=2$; in general, the conjecture remains open for $n\geq3$. It implies the complex Jacobian conjecture.