  We prove that a recently derived correlation equality between conserved
charges and their associated conserved currents for quantum systems far from
equilibrium [O.A. Castro-Alvaredo et al., Phys. Rev. X \textbf{6}, 041065
(2016)], is valid under more general conditions than assumed so far. Similar
correlation identities, which in generalized Gibbs ensembles give rise to a
current symmetry somewhat reminiscent of the Onsager relations, turn out to
hold also in the absence of translation invariance, for lattice models, and in
any space dimension, and to imply a symmetry of the non-equilibrium linear
response functions.
