An extension of Franklin's Theorem

Back in 1922, Franklin proved that every $3$-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.
It is well-known that each $3$-polytope has a vertex of degree at most $5$, called minor vertex. A $3$-path $uvw$ is an $(i,j,k)$-path if $d(u)\le i$, $d(v)\le j$, and $d(w)\le k$, where $d(x)$ is the degree of a vertex $x$. A $3$-path is minor $3$-path if its central vertex is minor.
The purpose of this note is to extend Franklin' Theorem to the $3$-polytopes with minimum degree at least $4$ by proving that there exist precisely the following ten tight descriptions of minor $3$-paths:$\{(6,5,6),(4,4,9),(6,4,8),(7,4,7)\}$, $\{(6,5,6),(4,4,9),(7,4,8)\}$, $\{(6,5,6),(6,4,9),(7,4,7)\}$, $\{(6,5,6),(7,4,9)\}$, $\{(6,5,8),(4,4,9),(7,4,7)\}$,$\{(6,5,9),(7,4,7)\}$, $\{(7,5,7),(4,4,9),(6,4,8)\}$, $\{(7,5,7),(6,4,9)\}$,$\{(7,5,8),(4,4,9)\}$, and $\{(7,5,9)\}$.