Three-involution bound for the first length parameter in dihedral groups

Let $D_n$ be a dihedral group with an order $n$ cyclic subgroup, and let $S$ be a generating set composed of three involutions, none of which belongs to that cyclic subgroup. Write $\lambda_1(D_n,S)$ for the first length parameter associated with $(D_n,S)$. Three-involution bound. We conjecture that

$$\lambda_1(D_n,S) \leq \left\lfloor \frac{n}{2} \right\rfloor + 1.$$

A complete proof is described as currently elusive; the theorem presented afterward supplies strong evidence for the bound in particular three-involution generating sets.