Bound on involution length in finite Coxeter groups assuming the Ancestor Conjecture

Let $W$ be a finite Coxeter group of rank $r$, and suppose the Ancestor Conjecture holds, so that every element has a unique ancestor decomposition. For $w\in W$, let $\ell_{\mathrm{inv}}(w)$ be the number of involutions in this decomposition, with $\ell_{\mathrm{inv}}(1)=0$.

Bound on involution length. For every $w\in W$,

$$\ell_{\mathrm{inv}}(w)\leq r.$$

This is a conditional consequence stated after the conjecture rather than an independent conjecture; the source gives no proof or resolution beyond the stated implication.