  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.
