Measure preserving maps with bounded total variation

Consider a piecewise affine Lipschitz map $\phi :\mathrm{\Omega }\to \mathbb{R}$, where $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is an open set, and assume that $x\mapsto x+t\mathrm{\nabla }\phi (x)$ is injective for almost every $t>0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., $\mathbf{\text{7}}$(4), 2025) the authors conjecture that every such $\phi$ must be locally convex. We prove the result assuming additionally $\mathrm{\nabla }\phi \in B{V}_{loc}(\mathrm{\Omega })$, for a more general class of measure preserving maps.