Concurrent normals of immersed manifolds

It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on boundary $K$. The conjecture known be true $n=2,3,4$. Motivated by recent results Y. Martinez-Maure, and an approach A. Grebennikov G. Panina, we prove following: Let compact smooth $m$-dimensional manifold $M^m$ immersed $ \mathbb{R}^n$. We assume one homology groups $H_k(M^m,\mathbb{Z}_2)$ with $k<m$ vanishes. Then under mild conditions, almost every normal line contains intersection $\beta +4$ $M^m$, where $\beta$ sum Betti numbers $M^m$.