Equitable coloring conjecture for outerplanar graphs with four or five colors

Let $G$ be an outerplanar graph, let $s obreak\text{ be }4$ or $5$, and for each vertex $v$ let $b1_v(G)$ denote the maximum size of an independent set containing $v$. An equitable $s$-coloring is a proper coloring with $s$ color classes whose sizes differ by at most one.

Equitable coloring conjecture. If

$$\min_{v\in V(G)}\alpha_v(G)\ge \left\lfloor\frac{|G|}{s}\right\rfloor,$$

then $G$ has an equitable $s$-coloring.

The condition is necessary for an equitable $s$-coloring, and the conjecture would extend the corresponding theorem for trees to all outerplanar graphs. It is known for $s\ge 6$, while the cases $s=4$ and $s=5$ remain open; the paper gives modest support by showing that the case $s=4$ implies the case $s=5$.