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We agree that $\rho_e = \rho_R + \rho_L = e \sum_I \left( \psi^\dagger_{R,I} \psi_{R,I} + \psi^\dagger_{L,I} \psi_{L,I} \right)$.  Then the continuity equation implies  $$ j_e(x) = \frac{e}{\pi} \sum_{I=1}^N \sum_{J=1}^{2N} \left[ V_{I,J} + -  V_{N+I,J} \right]\eta_{JJ}  \partial_x \phi_J(x) $$or equivalently  $$ j_e(x) = \frac{e}{\pi} \sum_{I,J=1}^N \left[ (V^R_{IJ}+V^{LR}_{N+I,J}) \partial_x \phi_J - (V^L_{N+I,N+J} + V^{RL}_{I,N+J}) \partial_x \phi_{N+J} \right] $$