The rank-two isometry conjecture for stable commutator length

Let $F$ be a free group, and let $\varphi:F_2\to F$ be an injective homomorphism from the free group of rank $2$. An injective homomorphism $\varphi$ is an isometry of stable commutator length if

Rank-two isometry conjecture. For every $g\in F_2$, one has

$$\operatorname{scl}_F(\varphi(g))=\operatorname{scl}_{F_2}(g).$$

The paper reports extensive computer evidence and some theoretical support for this claim. It would characterize every injective homomorphism from $F_2$ as preserving stable commutator length, although injective homomorphisms from free groups of rank at least $3$ need not be isometries.