  The very recent analysis by BaBar Collaboration indicates that the $X(3872)$
may favor the quantum number $J^{PC}=2^{-+}$ rather than the previously assumed
$1^{++}$. By pretending the $\eta_{c2}(1D)$ charmonium to be the $X(3872)$, we
study the parity-even radiative transition processes $\eta_{c2}(1D)\to
J/\psi(\psi')+\gamma$ within several phenomenological potential models. We take
the ${}^3D_1$ admixture in $\psi'$ into account, and consider the contributions
from the magnetic dipole ($M1$), electric quadrupole ($E2$), and magnetic
octupole ($M3$) amplitudes. It is found that the ratio of the branching
fractions of these two channels, as well as the absolute branching fraction of
$\eta_{c2}\to \psi'\gamma$, are in stark contradiction with the existing
\textsc{BaBar} measurements. This may indicate that the $2^{-+}$ assignment for
the $X(3872)$ is highly problematic.
