  We prove an $\varepsilon$-regularity theorem for $BV^\mathcal{B}$ minimizers
of strongly $\mathcal{B}$-quasiconvex functionals with linear growth, where
$\mathcal{B}$ is an elliptic operator of the first order. This generalises to
the $BV^\mathcal{B}$ setting the analogous result for $BV$ functions by F.
Gmeineder and J. Kristensen [Arch. Rational Mech. Anal. 232 (2019)]. The
results of this work cannot be directly derived from the $\mathcal{B} =\nabla$
case essentially because of Ornstein's "non-inequality". This adaptation
requires an abstract local Poincar\'e inequality and a fine Fubini-type
property to avoid the use of trace theorems, which in general fail when
$\mathcal{B}$ is elliptic.
