Each simplex in ${\mathbb{R}}^4$ has a point with at least $14$ emanating normals to its boundary

We prove that each simplex in ${\mathbb{R}}^4$ contains a point with at least $14$ emanating normals to the boundary. Earlier, the authors proved that each generic simple polytope in ${\mathbb{R}}^4$ contains a point with at least $12$ emanating normals to the boundary. This problem belongs to a piecewise linear counterpart of a longstanding problem concerning normals to smooth convex bodies.