 Solution

Yes, for all three ideal lattices under the stated assumptions. Cubic symmetry and equal rates make $\mathbf{\sigma }=\sigma I$ with $\sigma >0$, so $\mathbf{j}=\sigma \mathcal{E}$. Positive ions drift along the field. The current coefficient contains $Q^2$, so negative charge carriers would also produce a conventional current along the field, although their particle drift would reverse. Anisotropic barriers or a field strong enough to invalidate linear response would require a new calculation.