Equitable chromatic number conjecture for Pancake graphs

For each integer $n\geqslant 2$, let $P_n$ be the Pancake graph, let $\chi(P_n)$ denote its chromatic number, and let $\chi_{=}(P_n)$ denote its equitable chromatic number, the least number of colors in a proper coloring whose color-class sizes differ pairwise by at most one.

Pancake graph equitable coloring conjecture. For every $n\geqslant 2$,

$$\chi(P_n)=\chi_{=}(P_n).$$

Since equitable colorings are proper colorings with an additional balance condition, one always has $\chi(P_n)\leqslant\chi_{=}(P_n)$. The paper notes that the equality holds for the optimal colorings known there, while the assertion for all $n\geqslant2$ remains conjectural.