Exponential word-length growth with Whitehead complexity

Let $F_r$ be a free group, let $WC(w)$ be the Whitehead complexity of $w$, and define

$$W_m=\{w\in F_r\mid WC(w)=m\},$$ $$W_{m,c_r}=\{w\in W_m\mid |w|\geq c_r^m\}.$$

Exponential word-length growth conjecture. There exists a constant $c_r>1$ such that

$$\lim_{m\rightarrow\infty}\frac{|W_{m,c_r}|}{|W_m|}=1,$$

and the convergence is exponentially fast. This would formalize the observed exponential relationship between word length and Whitehead complexity. The source gives no resolution status.