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We present current bounds on SMEFT operators that are mainly constrained by Higgs-boson observables, under different assumptions for the flavour structure of the UV theory. We investigate how the accuracy reached through a dedicated Higgs-boson precision physics program is starting to have a major impact in setting a lower bound on the scale of new physics, and we discuss the relevance of considering the scale evolution of the SMEFT coefficients in this context. We compare our results with the literature, pointing out the consistency of the results in spite of the different assumptions adopted in each analysis, and we discuss future steps aimed at improving the accuracy of the fit.

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