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We consider the effect of $\alpha$ variation on the vibrational spectrum of Sr$_2$ in the context of a planned experiment to test the stability of $\mu\equiv m_e/m_p$ using optically trapped Sr$_2$ molecules [Zelevinsky et al., Phys. Rev. Lett. {\bf 100}, 043201; Kotochigova et al., Phys. Rev. A {\bf 79}, 012504]. …