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 $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ and any $u\in \mathrm{B}\mathrm{V}(\mathrm{\Omega },\mathcal{N})$, we show that $u$ has a lifting $v\in \mathrm{B}\mathrm{V}(\mathrm{\Omega },\mathcal{E})$. Our result proves a conjecture by Bethuel and Chiron.