Generic monotonicity conjecture for restricted Whitehead descent

Let $F_r$ be a free group, and let $T=T_r$ be the restricted set of Whitehead automorphisms described in the source. For each length $l$, let $NMin_l\subset F_r$ be the set of non-minimal elements of length $l$, and let $NMin_{l,T}$ be the subset on which Whitehead descent with respect to $T$ is monotone, meaning that it encounters no local minimum. Generic monotonicity conjecture.

$$\lim_{m\rightarrow\infty}\frac{|NMin_{l,T}|}{|NMin_l|}=1,$$

with exponentially fast convergence. This conjecture would imply that, on most non-minimal inputs, restricted Whitehead descent has a complexity bounded by a polynomial factor in $r$ times $WC(w)|w|$. The source gives no resolution status.