On regular homotopy of branched coverings of the sphere

Let Mⁿ be an orientable closed n-manifold and f, g: MⁿSⁿ branched coverings of the n-sphere Sⁿ. It is a theorem of H. Hopf that if f and g have the same degree then f and g are homotopic. Our interest is to find out whether f and g are then also regular homotopic, that is to say whether there is a level preserving branched covering H: Mⁿ×ISⁿ×I such that H₀=f and H₁=g. If n=2 or if n=3 and M³ is homeomorphic to S³ the answer to this question is affirmative. For some M³ not homeomorphic to S³ there are however counterexamples.