$B_s$-$\bar B_s$ mixing interplay with $B$ anomalies

After reviewing the theoretical uncertainties entering the Standard Model determination of the mass difference of the neutral $B_s$-$\bar B_s$ meson system, $\Delta M_s^{\rm SM}$, we discuss the implications of its updated value for new physics models addressing the experimental anomalies in semi-leptonic $B$ decays. Using the most recent FLAG average of lattice results for the non-perturbative matrix elements and the CKM-fitter determination of $V_{cb}$ points to a $1.8 \, \sigma$ discrepancy in $\Delta M_s^{\rm SM} > \Delta M_s^{\rm exp}$. Extending the analysis in Ref. [1] we show that the latter tension cannot be easily accommodated within single mediator models, whenever the same mediator is also responsible for the $b \to s \ell \ell$ anomalies.