Average length-reduction complexity conjecture for Whitehead descent

Let $F=F_n$ be a free group of rank $n$, and let $NMin_l\subset F$ be the set of all non-minimal elements of length $l$. Let $LR(w)$ denote the length-reducing complexity associated with Whitehead descent. Average length-reduction complexity conjecture. There is a constant $LR_n$ such that

$$\limsup_{l\rightarrow\infty}\frac{1}{|NMin_l|}\sum_{w\in NMin_l}|LR(w)|=LR_n.$$

This conjecture predicts a rank-dependent limiting upper-average number of length-reducing steps for non-minimal words, and is intended to explain the observed average complexity of the standard Whitehead descent algorithm. The source gives no resolution status.