The rationality conjecture for maximal rotation numbers in the commutator subgroup

Let $F$ be the free group generated by $a$ and $b$, and let $w\in [F,F]$. Commutator-subgroup rationality conjecture. Then

$$\max_{r,s} R(w,r,s)\in\mathbb Q.$$

This is a rationality conjecture for the maximum of $R(w,r,s)$ over the rotation parameters when the word lies in the commutator subgroup. It is stated as part of the paper's program on rationality of extremal rotation numbers and is open in the supplied text.