Is the $J^P=2^{-}$ assignment for the $X(3872)$ compatible with the radiative transition data?

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 ${}^3{D}_1$ admixture in ${\psi }^{\mathrm{\prime }}$ 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 ${\eta }_{c2}\to {\psi }^{\mathrm{\prime }}\gamma$, are in stark contradiction with the existing $\text{textsc}BaBar$ measurements. This may indicate that the $2^{-+}$ assignment for the $X(3872)$ is highly problematic.