Subexponential growth conjecture for equal-length Whitehead orbit elements

Let $F_r$ be a free group, and let $w\in F_r$ be a word whose length cannot be reduced by any elementary Whitehead automorphism; in particular, $w$ is cyclically reduced. Consider the elements in the automorphic orbit of $w$ having the same length as $w$. Subexponential equal-length orbit conjecture. The number of such elements is subexponential in $|w|$. This conjecture would imply that, for arbitrary rank $r>2$, the average-case time complexity of the Whitehead algorithm is linear, extending the known rank-two result. The paper describes the required orbit-growth estimate as an open problem.