Benedetti–Shiota conjecture on real algebraic links

Let $f:\mathbb{R}^4\to\mathbb{R}^2$ be a polynomial map with an isolated singularity at the origin, meaning that $f(0)=(0,0)$, its first derivatives vanish at the origin, and its Jacobian has full rank away from the origin in some neighbourhood. For sufficiently small $\rho>0$, its zero set intersects

$$S^3_{\rho}=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:\sum_{i=1}^4 x_i^2=\rho^2\}$$

in the link of the singularity. A link $L$ is real algebraic if it arises in this way. Benedetti–Shiota conjecture. A link $L$ is real algebraic if and only if it is fibered.

Milnor's theorem gives the forward implication: every real algebraic link is fibered. The conjecture asserts the converse, which is not established in the supplied text.