Tight description of 4-paths in 3-polytopes with minimum degree 5

Back in 1922, Franklin proved that every 3-polytope P5 with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions. In particular, Jendrol' and Madaras (1996) ensured a 4-path with the vertex degree-sum at most 23. The purpose of this note is to prove that every P5 has a (5, 6, 6, 6)-path or (5, 5, 5, 7)-path, where all parameters are tight.