Schur property for jump parts of gradient measures

We consider weakly null sequences in the Banach space of functions of bounded variation $\mathrm{B}\mathrm{V}({\mathbb{R}}^d)$. We prove that for any such sequence $\{f_n\}$ the jump parts of the gradients of functions $f_n$ tend to $0$ strongly as measures. It implies that Dunford–Pettis property for the space $\mathrm{S}\mathrm{B}\mathrm{V}$ is equivalent to the Dunford–Pettis property for the Sobolev space $W^{1,1}\mathrm{.}$