Measure preserving maps with bounded total variation

Consider a piecewise affine Lipschitz map $\phi : \Omega \to \mathbb R$, where $\Omega \subset \mathbb R^d$ is an open set, and assume that $x \mapsto x + t \nabla \phi(x)$ 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., \textbf{7}(4), 2025) the authors conjecture that every such $\phi$ must be locally convex. We prove the result assuming additionally $\nabla \phi \in BV_{loc}(\Omega)$, for a more general class of measure preserving maps.