Under the same assumptions as the local error lemma, there exists a constant $C = C(p, \gamma)$ such that
\begin{align*}
\|y_T - y^{\mathrm{Euler};\, \lfloor\gamma\rfloor,\, \mathcal{D}}_T\| \leq C\, e^{C\,\omega(0,T)} \sum_{i=1}^n \omega(t_{i-1}, t_i)^{(\lfloor\gamma\rfloor + 1)/p},
\end{align*}
where the control function is $\omega(s, t) = \|f\|^p_{\mathrm{Lip}^\gamma}\, \|x\|^p_{p\text{-var};[s,t]}$.