  A conjugate skew gain graph is a skew gain graph with the labels (also
called, the conjugate skew gains) from the field of complex numbes on the
oriented edges such that they get conjugated when we reverse the orientation.
In this paper we introduce distance matrices for conjugate skew gain graphs and
characterize balanced conjugate skew gain graphs using these matrices. We
provide explicit formulae for the distance spectra of certain conjugate skew
gain graphs.
