  We have studied the contribution of the state $X(3930)$, coming from the
interaction of the $D \overline{D}$ and $D^{+}_s D^{-}_s$ channels, to the $B^-
\to K^- J/\psi \omega $ decay. The purpose of this work is to offer a
complementary tool to see if the $X(3930)$ state observed in the $D^+ D^-$
channel is the same or not as the $X(3960)$ resonance claimed by the LHCb
collaboration from a peak in the $D^{+}_s D^{-}_s$ mass distribution around
threshold. We present results for what we expect in the $J/\psi \omega $ mass
distribution in the $B^- \to K^- J/\psi \omega $ decay and conclude that a
clear signal should be seen around $3930\,\rm MeV$. At the same time, finding
no extra resonance signal at $3960\,\rm MeV$ would be a clear indication that
there is not a new state at $3960\,\rm MeV$, supporting the hypothesis that the
near-threshold peaking structure peak in the $D^{+}_s D^{-}_s$ mass
distribution is only a manifestation of a resonance below threshold.
