Recent results by the Fermilab Lattice and MILC collaborations on the hadronic matrix elements entering $B_{s,d}-\bar B_{s,d}$ mixing have reached an unprecedented precision. Interestingly the Standard Model (SM) predictions using these updated values, together with the CKM elements obtained from tree-level decays, exhibit a significant tension with the measured values of the mass differences $\Delta M_s$, $\Delta M_d$ and their ratio. Assessing this tension in a model-independent way, it can be shown that models with Constrained Minimal Flavour Violation can not improve the situation with respect to the SM, in particular when the correlation with the CP-violating parameter $\varepsilon_K$ is taken into account. The new lattice results, if eventually confirmed by independent calculations, therefore imply the presence of new sources of flavour violation in the $\Delta F=2$ sector.