Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

For a finite set $A \subseteq \mathbb{R}^n$, consider a function $u \in \mathrm{BV}_{\mathrm{loc}}^2(\mathbb{R}^n)$ such that $\nabla u \in A$ almost everywhere. If $A$ is convex independent, then it follows that $u$ is piecewise affine away from a closed, countably $\mathcal{H}^{n - 1}$-rectifiable set. If $A$ is affinely independent, then $u$ is piecewise affine away from a closed $\mathcal{H}^{n - 1}$-null set.