Describing $4$-paths in $3$-polytopes with minimum degree $5$

Back in 1922, Franklin proved that each $3$-polytope 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 degree-sum at most $23$. The purpose of this note is to prove that each 3-polytope with minimum degree $5$ has a $(6,5,6,6)$-path or $(5,5,5,7)$-path, which is tight and refines both above mentioned results.