We study locally conformal calibrated $G_2$-structures whose underlying Riemannian metric is Einstein, showing an integral formula for compact manifolds. We show that a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated $G_2$-structure unless the underlying metric is flat. In contrast to the compact case we provide a non-compact example of homogeneous manifold endowed with a locally conformal calibrated $G_2$-structure whose associated Riemannian metric is Einstein and non Ricci-flat. The homogeneous Einstein metric is a rank-one extension of a Ricci soliton on the 3-dimensional complex Heisenberg group endowed with a left-invariant coupled half-flat SU(3)-structure $(\omega, \psi)$ such that $d \omega = - Re(\psi)$.
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