Describing neighborhoods of $5$-vertices in a class of $3$-polytopes with minimum degree $5$

Lebesgue proved in 1940 that each $3$-polytope with minimum degree $5$ contains a $5$-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences
$$ \begin{gathered} (6,6,7,7,7),\ (6,6,6,7,9),\ (6,6,6,6,11),\\ (5,6,7,7,8),\ (5,6,6,7,12),\ (5,6,6,8,10),\ (5,6,6,6,17),\\ (5,5,7,7,13),\ (5,5,7,8,10),\ (5,5,6,7,27),\ (5,5,6,6,\infty),\ (5,5,6,8,15),\ (5,5,6,9,11),\\ (5,5,5,7,41),\ (5,5,5,8,23),\ (5,5,5,9,17),\ (5,5,5,10,14),\ (5,5,5,11,13). \end{gathered} $$
We prove that each $3$-polytope with minimum degree $5$ without vertices of degree from $7$ to $10$ contains a $5$-vertex whose set of degrees of its neighbors is majorized by one of the following sequences: $(5,6,6,5,\infty)$, $(5,6,6,6,15)$, and $(6,6,6,6,6)$, where all parameters are tight.