Asymptotic Nielsen-reduction conjecture for non-minimal free-group elements

Let $F$ be a free group. For each positive integer $n$, let $U_n$ be the set of all non-minimal elements in $F$ of length $n$, and let $NU_n\subset U_n$ be the subset consisting of elements that have length-reducing Nielsen automorphisms. Asymptotic Nielsen-reduction conjecture.

$$\lim_{n \rightarrow \infty} \frac{|NU_n|}{|U_n|}=1.$$

The claim formalizes the experimental observation that, with probability tending to one as length grows, a non-minimal element admits a length-reducing Nielsen automorphism. The source reports numerical evidence for free groups of ranks $3$, $4$, and $5$, but gives no proof or resolution.