The eventual rotation-quasimorphism formula for scl on a punctured torus

Let $F_2$ be the free group on generators $a,b$, and let $w\in [F_2,F_2]$ be arbitrary. Let $S$ be a hyperbolic once-punctured torus with $\pi_1(S)\cong F_2$, and let $\operatorname{rot}_S$ be the associated rotation quasimorphism. Eventual rotation-quasimorphism conjecture. For sufficiently large integers $n$, there is an equality

$$\operatorname{scl}(w[a,b]^n)=\frac{\operatorname{rot}_S(w[a,b]^n)}{2}.$$

Computer experiments suggest that the corresponding geodesic rationally bounds an immersed surface for sufficiently large $n$, but a general argument is lacking; the conjecture gives a precise formula for stable commutator length in this family.