Concurrent normals of immersed manifolds

It is conjectured since long that for any convex body K ⊂ Rⁿ there exists a point in the interior of K which belongs to at least 2n normals from different points on the boundary of K. The conjecture is known to be true for n = 2, 3, 4. Motivated by a recent results of Y. Martinez-Maure, and an approach by A. Grebennikov and G. Panina, we prove the following: Let a compact smooth m-dimensional manifold M^m be immersed in Rⁿ. We assume that at least one of the homology groups H_k (M^m, Z₂) with k < m vanishes. Then under mild conditions, almost every normal line to M^m contains an intersection point of at least β + 4 normals from different points of M^m, where β is the sum of Betti numbers of M^m.