Energy from Ellwood invariant for solutions involving $X^0$ variables

For some classical solutions ${\mathrm{\Psi }}_{\mathrm{s}\mathrm{o}\mathrm{l}}$ in Witten’s bosonic string field theory, it was proven that energy of the solution is proportional to the Ellwood invariant $\mathrm{T}\mathrm{r}(\mathcal{V}{\mathrm{\Psi }}_{\mathrm{s}\mathrm{o}\mathrm{l}})$ with $\mathcal{V}=c\bar{c}\mathrm{\partial }{X}^0\bar{\mathrm{\partial }}{X}^0$. We examine the relation for solutions involving $X^0$ variables. As a result, we obtain that the relation may not hold for such solutions. Namely, there is a possibility that the energy is not proportional to the Ellwood invariant.