On the monotone upper bound problem

The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n) M ubt (d, n) provided by McMullen's (1970) Upper Bound Theorem is tight, where M ubt (d, n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d,n) M ubt (d, n) holds with equality for small dimensions (