The surjectivity conjecture for Jacobian operators of polynomial mappings

Let $F=(f_1,\ldots,f_n):\mathbb R^n\to\mathbb R^n$ be a polynomial mapping with $J(F)$ its Jacobian determinant and $S_F$ its non-properness set. For $i\in\{1,\ldots,n\}$, define

$$\Delta_i^F(g)=J(f_1,\ldots,f_{i-1},g,f_{i+1},\ldots,f_n).$$

Surjectivity conjecture. If $J(F)$ is nowhere vanishing and $\operatorname{codim} S_F\geq2$, then

$$\Delta_i^F(C^{\infty}(\mathbb R^n))=C^{\infty}(\mathbb R^n)$$

for $n-1$ indices $i\in\{1,\ldots,n\}$. The conjecture is related to Jelonek's conjecture and to the paper's global injectivity results; no resolution is given in the supplied text.