Combinatorial structure of faces in triangulated $3$-polytopes with minimum degree $4$

In 1940, Lebesgue proved that every $3$-polytope with minimum degree at least $4$ contains a $3$-face for which the set of degrees of its vertices is majorized by one of the entries: $(4,4,\infty)$, $(4,5,19)$, $(4,6,11)$, $(4,7,9)$, $(5,5,9)$, and $(5,6,7)$. This description was strengthened by Borodin (2002) to $(4,4,\infty)$, $(4,5,17)$, $(4,6,11)$, $(4,7,8)$, $(5,5,8)$, and $(5,6,6)$.
For triangulations with minimum degree at least $4$, Jendrol' (1999) gave a description of faces: $(4,4,\infty)$, $(4,5,13)$, $(4,6,17)$, $(4,7,8)$, $(5,5,7)$, and $(5,6,6)$.
We obtain the following description of faces in plane triangulations (in particular, for triangulated $3$-polytopes) with minimum degree at least $4$ in which all parameters are best possible and are attained independently of the others: $(4,4,\infty)$, $(4,5,11)$, $(4,6,10)$, $(4,7,7)$, $(5,5,7)$, and $(5,6,6)$.
In particular, we disprove a conjecture by Jendrol' (1999) on the combinatorial structure of faces in triangulated $3$-polytopes.