Ambrosio–De Lellis–Mantegazza relaxation conjecture for line energies

Let $\mathcal{I}_f$ be a line energy defined on $BV$, let $\overline{\mathcal{I}_f}$ be its relaxation in $L^1$,

$$\overline{\mathcal{I}_f}(m)=\operatorname{Inf}\left\{\liminf_{n\to+\infty}\mathcal{I}_f(m_n):m_n\in BV\text{ and }m_n\to m\text{ in }L^1\right\}.$$

Ambrosio–De Lellis–Mantegazza conjecture. If $f(t)=t^p$ with $1\leq p\leq3$, then $\overline{\mathcal{I}_f}$ is lower semicontinuous for the strong topology in $L^1$.

The conjecture concerns lower semicontinuity after relaxation, a central issue in the variational theory of line energies. The source provides no resolution evidence, so the conjecture is recorded as open.