The degree bound for compositions of automorphisms of free groups

Let $F_n$ be the free group of rank $n$, and let $f,g$ be automorphisms of $F_n$. Write $\operatorname{deg}(f)$ and $\operatorname{deg}(g)$ for their degrees. Degree-composition conjecture.

$$\operatorname{deg}(f\circ g)\leq (n+\operatorname{deg}(f))(n+\operatorname{deg}(g)).$$

This conjecture proposes a general upper bound for the degree of a composition in terms of the rank of the free group and the degrees of the two factors; the supplied source gives no information about its resolution.