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$$ \phi_{N+I} = - \phi_L^{(G))} $$  so finally  I conclude that (dropping the factor of K): that:  $$ \rho_R(x) = \frac{1}{2\pi} \partial_x \phi_R^{(G))} =  \frac{-1}{2\pi} \partial_x \phi_R $$ $$ \rho_L(x) = \frac{-1}{2\pi} \partial_x \phi_L^{(G))} = \frac{-1}{2\pi} \partial_x  \phi_R $$ From the commutation for the non-chiral boson.