Milnor's product conjecture for fibers of composed polynomial mappings

Let $\psi=(f_1,\ldots,f_k):\mathbb{R}^n\to\mathbb{R}^k$ be a polynomial mapping with an isolated singularity at the origin. Compose $\psi$ with the projection $\mathbb{R}^k\to\mathbb{R}^{k-1}$ to obtain a new mapping $\mathbb{R}^n\to\mathbb{R}^{k-1}$, also with an isolated singularity at the origin. Milnor's product conjecture. The fiber of the fibration associated with this new mapping is homeomorphic to the product of the old fiber with the unit interval.