Five-color version of the planar equitable coloring theorem

Let $G$ be a planar graph, and let $w_0,w_1$ be its two vertices with largest degrees, breaking ties arbitrarily. Let $s\ge 40$. If there exist disjoint independent sets $I_0,I_1$ such that

$$|I_0|=\left\lfloor\frac{n}{s}\right\rfloor, \qquad |I_1|=\left\lfloor\frac{n+1}{s}\right\rfloor,$$

and $w_0,w_1\in I_0\cup I_1$, then $G$ has an equitable $s$-coloring.

Five-color planar theorem conjecture. The same statement remains true when the lower bound $s\ge 40$ is replaced by $s\ge 5$.

The paper proves the planar theorem for $s\ge 40$ and conjectures that the threshold can be reduced to $5$. The conjecture concerns strengthening the sufficient condition for equitable colorability of planar graphs; the authors note that they have not optimized the proved bound and believe it is far from best possible.