Equality of the bifurcation and asymptotic sets at infinity

Let $f\colon \mathbb{R}^n\to\mathbb{R}$ be a real polynomial. Denote by $\mathcal{B}_{\infty}(f)$ the bifurcation set at infinity and by $\mathcal{S}_{\infty}(f)$ the set of asymptotic critical values at infinity. The equality conjecture. One has

$$\mathcal{B}_{\infty}(f)=\mathcal{S}_{\infty}(f).$$

The conjecture proposes that the bifurcation values at infinity are exactly the asymptotic critical values at infinity. The preceding discussion gives the inclusion $\mathcal{S}_{\infty}(f)\subset K_{\infty}(f)$ and exhibits examples where this inclusion is strict, while a related conjecture from the literature has a real counterexample. The status of the equality itself is not resolved in the supplied text.