Heights of minor faces in triangle-free $3$-polytopes

The height $h(f)$ of a face $f$ in a $3$-polytope is the maximum of the degrees of vertices incident with $f$. A $4$-face is pyramidal if it is incident with at least three $3$-vertices. We note that in the $(3,3,3,n)$-Archimedean solid each face $f$ is pyramidal and satisfies $h(f)=n$.
In 1940, Lebesgue proved that every quadrangulated $3$-polytope without pyramidal faces has a face $f$ with $h(f)\le11$. In 1995, this bound was improved to $10$ by Avgustinovich and Borodin. Recently, the authors improved it to $8$ and constructed a quadrangulated $3$-polytope without pyramidal faces satisfying $h(f)\ge8$ for each $f$.
The purpose of this paper is to prove that each $3$-polytope without triangles and pyramidal $4$-faces has either a $4$-face with $h(f)\le10$ or a $5$-face with $h(f)\le5$, where the bounds $10$ and $5$ are sharp.