Confused about selection rules in optical transitions

[Malamala]

Hello! This question is motivated by this paper in particular (let me know if it is not open access). They measure transitions between electronic levels inside an ion, placed inside a crystal. In Fig. 2, they show measured transitions between 2 electronic levels, one of them having $J = 15/2$ and $M_J=\pm 5/2$ while the other one has $J = 9/2$ and $M_J=\pm 9/2$ which means that the transitions have $\Delta J = 3$ and, for the largest difference, $\Delta M_J = 9/2 - (-5/2) = 7$. How is this possible? I understand that, $J$ and $M_J$ are not perfectly good quantum numbers due to the crystal field, but they are still quite good to a certain degree. Also, even if there is some mixing of other electronic states with different $J$ and $M_J$ values, I wouldn't expected states with so different quantum numbers to have a significant contribution to the mixing. Given that the paper claims to search for very weak axion signals, I assume the strategy is not to use highly forbidden transitions, on top of the fact that the signal sought for is very weak (but anyway, they seem to be able to easily excite these transitions with a normal laser). What am I missing here? Thank you!