all $s,t\in [0, T]$.

(ii) (Uniqueness and consistency) The lift $\mathbf{X}$ is unique in the sense that it is the $d_{p -v a r}$-limit in $L^{q} (\Omega)$, $q \in [1,\infty )$, of any sequence $S_{^{\left \lfloor p\right \rfloor }} (X^{\pi}) $ with $\left\| \pi\right\| \rightarrow 0$. Furthermore, if $X$ has a.s. sample paths of finite $[1 ,2)$-variation, $\mathbf{X}$ coincides with the signature of $X$.

Moreover, Proposition 17 in [@fv2010a] shows that for all $h\in\mathcal{H}^{d}$, $$\begin{aligned}
 \label{CMembedding}
\left\|  h\right\| _{\rho-var; [0, T]} \leq\left\| h\right\| _{\mathcal{H}^{d}} \sqrt{\left\|  R\right\| _{\rho-var; [0, T]^{2}}},\end{aligned}$$ which implies that $\mathcal{H}^{d} \hookrightarrow\mathcal{C}^{\rho-var}([0 ,T] ;\mathbb{R}^{d})$ whenever $R$ has finite 2D $\rho$-variation. Thus if $\rho\in\left[ 1, \frac{3}{2}\right) $, corresponding to $2 \leq p < 3$, we have complementary regularity between $X$ and any path in the Cameron-Martin space.

In the case $\rho\in\left[  \frac{3}{2}, 2 \right)$, there exists a recent result for complementary regularity between the Cameron-Martin paths and $X$ which requires the following definition.

We say that a function $f \in V^{(p, q)-var}([0, T]^{2})$ is of mixed (left) $(p, q)$-variation if $$\begin{aligned}
\sup_{\pi} \left(  \sum_{i} \left(  \sum_{j} \left|  f
\begin{pmatrix}
s_{i} & s_{i+1}\\
t_{j} & t_{j+1}\end{pmatrix}
\right| ^{p} \right) ^{\frac{q}{p}} \right) ^{\frac{1}{q}} < \infty,\end{aligned}$$ where the supremum runs over all 