Flip-length bound by the maximum rotation length in dihedral groups

Let $D_n$ be a dihedral group, let $S$ be a generating set, and let $l_S(g)$ denote the length of an element $g$ with respect to $S$. For an integer $m$ with $0<m<n$, consider the element $r^m f$. Flip-length bound. We conjecture that

$$l_S(r^m f) \leq \max_{0 \leq k < n}\{l_S(r^k)\}.$$

This conjecture follows the observation that bounds for powers of $r$ can be transferred to flips by expressing $r^m f$ using suitable generators. Its truth is stated not to affect the results of the lemmas that follow.