Growth-rate conjecture for cyclically reduced automorphic images

Let $F_r$ be a free group and let $w\in F_r$ be a non-primitive element. Let $CA(w)$ denote the set of cyclically reduced automorphic images of $w$. Cyclically reduced orbit growth conjecture. The growth rate of $CA(w)$ is less than $\sqrt{2r-1}$. This conjecture is presented as a quantitative form of the observation that non-primitive automorphic orbits are much thinner than the orbit of a primitive element. The paper states that it is weaker than the subexponential equal-length orbit conjecture and attributes it as related to work of Puder.