Existence of a function eliminating the asymptotic set when the bifurcation set is empty

Let $G=(G_1,\ldots,G_{n-1}):\mathbb{C}^n\to\mathbb{C}^{n-1}$ be a polynomial mapping with $n\geq 3$, and let $B(G)$ denote its bifurcation set. For a function $\rho$, let $\mathcal{S}_G(\rho)$ denote the associated set at infinity.

Existence conjecture. Does there exist a function $\rho$ such that

$$B(G)=\emptyset \quad\Longrightarrow\quad \mathcal{S}_G(\rho)=\emptyset?$$

The question asks whether an appropriate choice of the function $\rho$ can make the relevant set at infinity empty whenever the polynomial mapping has no bifurcation values. The surrounding discussion gives an example showing that the associated intersection homology behavior can depend on the choice of $\rho$; the existence of such a function is left as a conjectural question.