Light and low $5$-stars in normal plane maps with minimum degree $5$

It is known that there are normal plane maps (NPMs) with minimum degree $\delta=5$ such that the minimum degree-sum $w(S_5)$ of $5$-stars at $5$-vertices is arbitrarily large. The height of a $5$-star is the maximum degree of its vertices. Given an NPM with $\delta=5$, by $h(S_5)$ we denote the minimum height of a $5$-stars at $5$-vertices in it.
Lebesgue showed in 1940 that if an NPM with $\delta=5$ has no $4$-stars of cyclic type $(\overrightarrow{5,6,6,5})$ centered at $5$-vertices, then $w(S_5)<68$ and $h(S_5)<41$. Recently, Borodin, Ivanova, and Jensen lowered these bounds to $55$ and $28$, respectively, and gave a construction of a $(\overrightarrow{5,6,6,5})$-free NPM with $\delta=5$ having $w(S_5)=48$ and $h(S_5)=20$.
In this paper, we prove that $w(S_5)<51$ and $h(S_5)<23$ for each $(\overrightarrow{5,6,6,5})$-free NPM with $\delta=5$.