The BD jump-set conjecture

Let $u\in\operatorname{BD}_{\mathrm{loc}}(\mathbb{R}^d)$, with $S_u$ its approximate discontinuity set and $J_u$ its jump set. BD jump-set conjecture. For all $u\in\operatorname{BD}_{\mathrm{loc}}(\mathbb{R}^d)$,

$$\mathcal{H}^{d-1}(S_u\setminus J_u)=0.$$

This would establish the missing analogue of the corresponding structural statement for functions of bounded variation. It is one of the major open problems in the theory of BD-maps, and only partial results are currently known.