Is BV a Banach lattice with respect to a special cone?

Question

Let $BV$ be the space of functions on $[0,1]$ of bounded variations equipped with the standard norm $\Vert f\Vert := \Vert f\Vert_{L^1} + \mathrm{var}(f)$, where $\mathrm{var}(f)$ is the total variation of $f$. Consider a cone \begin{equation} \mathcal C:=\left\{ f \in BV : f \ge 0, \; \mathrm{var}( f ) \le \int f dx \right\} \end{equation} (this cone has a background in dynamical systems theory.) Introduce a partial order $\preceq$ on $BV$ by $f\preceq g$ iff $g-f \in \mathcal C$. Then, is $(BV,\Vert \cdot \Vert ,\preceq)$ a Banach lattice?