Lifting for manifold-valued maps of bounded variation

Let $\mathcal{N}$ be a smooth, compact, connected Riemannian manifold without boundary. Let $\mathcal{E}\to\mathcal{N}$ be the Riemannian universal covering of $\mathcal{N}$. For any bounded, smooth domain $\Omega\subseteq\mathbb{R}^d$ and any $u\in\mathrm{BV}(\Omega, \, \mathcal{N})$, we show that $u$ has a lifting $v\in\mathrm{BV}(\Omega, \, \mathcal{E})$. Our result proves a conjecture by Bethuel and Chiron.