The fine-property conjecture for functions in deviatoric bounded deformation

Let $\Omega$ be the domain, let $\mathrm{BD}_{\mathrm{dev}}(\Omega)$ denote the space of functions with deviatoric bounded deformation, and let $\mathcal{E}_d u$ be the deviatoric symmetric-gradient measure of $u$. Denote by $S_u$ the set of points where $u$ is not approximately continuous, and by $J_u$ its jump set.

Fine-property conjecture. For all $u\in \mathrm{BD}_{\mathrm{dev}}(\Omega)$, one has

$$|\mathcal{E}_d u|(S_u\setminus J_u)=0.$$

This conjecture asks whether the deviatoric symmetric-gradient measure charges the set of non-approximately-continuous points only on the jump set, as happens for functions in $\mathrm{BD}$ and $\mathrm{BV}$. It remains open because the usual slicing method does not generally produce one-dimensional $\mathrm{BV}$ slices for $\mathrm{BD}_{\mathrm{dev}}$ functions.