Is Total Angular Momentum Conserved in the Hydrogen Ion Molecule H₂⁺?

[Jamister]

in the book Molecular Physics of Wolfgang Demtroder, he claims for hydrogen ion molecule $H_2 ^+$ the total angular momentum in absolute value is a constant quantum number. I never saw it any other book and I think it's wrong, because there is no reason for it and he doesn't prove it. here is what he says in the book:

what do you think?

 

[TeethWhitener]

This all comes from Noether’s theorem. The nuclear potential in $H_2^+$ is not spherically symmetric, so the total angular momentum is not conserved. However, it is axially symmetric, so quantities associated with the axis of symmetry are conserved.