Let $x \in G\Omega_p(\mathbb{R}^d)$ be a geometric $p$-rough path and $f = (f_1, \ldots, f_d)$ a family of $\mathrm{Lip}^{\gamma-1}$ vector fields on $\mathbb{R}^e$ with $\gamma > p$. Let $y$ be a solution to the RDE. Then there exists a constant $C = C(p, \gamma)$ such that \begin{align*} \|y_t - \mathcal{E}_{s,t}^{\lfloor \gamma \rfloor}(y_s;\, f,\, x)\| \leq C\!\left(\|f\|_{\mathrm{Lip}^{\gamma-1}}\, \|x\|_{p\text{-var};[s,t]}\right)^\gamma. \end{align*}