We perform a theoretical study of the $chi_{cJ} to phi K^* bar K to phi Kpi bar K$ reaction taking into account the $K^* bar K$ final state interaction, which in the chiral unitary approach is responsible, together with its coupled channels, for the formation of the low lying axial vector mesons, in this case the $h_1(1380)$ given the selection of quantum numbers. Based on this picture we can easily explain why in the $chi_{c0}$ decay the $h_1(1380)$ resonance is not produced, and, in the case of $chi_{c1}$ and $chi_{c2}$ decay, why a dip in the $K^+ pi^0 K^-$ mass distribution appears in the 1550-1600 MeV region, that in our picture comes from a destructive interference between the tree level mechanism and the rescattering that generates the $h_1(1380)$ state. Such a dip is not reproduced in pictures where the nominal $h_1(1380)$ signal is added incoherently to a background, which provides support to the picture where the resonance appears from rescattering of vector-pseudoscalar components.
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