  We study the $G_2$ analogue of the Goldberg conjecture on non-compact
solvmanifolds. In contrast to the almost-K\"ahler case we prove that a
7-dimensional solvmanifold cannot admit any left-invariant calibrated
$G_2$-structure $\varphi$ such that the induced metric $g_{\varphi}$ is
Einstein, unless $g_{\varphi}$ is flat. We give an example of 7-dimensional
solvmanifold admitting a left-invariant calibrated $G_2$-structure $\varphi$
such that $g_{\varphi}$ is Ricci-soliton. Moreover, we show that a
7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any
left-invariant cocalibrated $G_2$-structure $\varphi$ such that the induced
metric $g_{\varphi} = g$.
