Heights of minor 5-stars in 3-polytopes with minimum degree 5 and no vertices of degree 6 and 7

We prove that every such polytope P∗ satisfies h(P∗)≤14, which bound is sharp. Moreover, if 6-vertices are allowed but 7-vertices forbidden, or vice versa, then the height of minor 5-stars in P5 under the absence of minor (5,5,5,5,∞)-stars can reach 15 or 17, respectively.