Existence of Milnor vector fields for real analytic map germs

Let $G:(\mathbb{R}^{m},0)\to (\mathbb{R}^{p},0)$, with $m>p\geq 2$, be an analytic nice map germ with radial discriminant such that $\Psi_G$ is $\rho$-regular. Suppose that the Milnor–Hamm tube and sphere fibrations exist for every sufficiently small $\varepsilon>0$ and $0<\eta\ll\varepsilon$.

Milnor vector field conjecture. If both fibrations exist under these conditions, then they are equivalent.

This conjecture concerns the general existence problem for Milnor vector fields and the equivalence of the Milnor–Hamm tube and sphere fibrations. The surrounding discussion states that sufficient conditions are known in several settings, but that no proof is known in complete generality and no counterexample has been found.