  We investigate the $J/\psi \phi$ invariant mass distribution of the
$e^+e^-\to \gamma J/\psi\phi$ reaction at a center of mass energy of
$\sqrt{s}=4.6$~GeV measured by the BESIII collaboration, who concludes that no
significant signals are observed for $e^+e^- \to \gamma X(4140)$ because of the
low statistics. We show, however, that the three bump structures in the $J/\psi
\phi$ invariant mass distribution, though given only by three data points, are
compatible with the existence of the $X(4140)$ state, appearing as a peak, and
a strong cusp structure at the $D^*_s\bar{D}^*_s$ threshold, resulting from the
molecular nature of the $X(4160)$ state, which also provides a substantial
contribution to the reaction. This is consistent with our previous analysis on
the $B^+\to J/\psi\phi K^+$ decay measured by the LHCb collaboration.
  We strongly call for the measurement of this process with more statistics to
further pin down the nature of the $X(4140)$ and $X(4160)$ resonances.
