Let $\mathbb{S}^{1}$ be the unit circle, $\Omega $ a smooth, bounded and simply connected domain in $\mathbb{R}^{2}$, and $k$ a positive integer. We prove that the set of configurations $a=\left( a_{1},...,a_{k}\right) \in \Omega ^{k}$ for which each $u\in W^{1,1}\left( \Omega ,\mathbb{S}^{1}\right) \cap C(\Omega \backslash \left\{ a_{1},...,a_{k}\right\} )$ admits a unique ($mod\,2\pi $) minimal $BV$-lifting $\varphi\in BV (\Omega, \mathbb{R})$ is of full measure in $\Omega ^{k}$. In particular, this implies that the set of those $ u\in W^{1,1}\left( \Omega,\mathbb{S}^{1}\right)$ that admit a unique ($mod\, 2\pi $) minimal $BV$-lifting is dense in $W^{1,1}$ $\left(\Omega,\mathbb{S}^{1}\right)$. This answers a question of Brezis and Mironescu.