Large $h\to b s$ in generic two-Higgs-doublet models

We investigate the possible size of $h\to bs$ in two-Higgs-doublet models with generic Yukawa couplings. Even though the corresponding rates are in general expected to be small due to the indirect constraints from $B_s\to\mu^{+}\mu^{-}$ and $B_s$--$\overline{B}_s$ mixing, we find regions in parameter space where $h\to bs$ can have a sizable branching ratio well above 10\%. This requires a tuning of the neutral scalar masses and their couplings to muons, but then all additional constraints such as $B\to X_s \gamma$, $(g-2)_\mu$, and $h\to\mu^{+}\mu^{-}$ are satisfied. In this case, $h\to bs$ can be a relevant background in $h\to b\bar{b}$ searches and vice versa due to the imperfect $b$-tagging purity. Furthermore, if $h\to bs$ is sizeable, one expects two more scalar resonances in the proximity of $m_h$. We briefly comment on other flavour violating Higgs decays and on the 95 GeV $\gamma\gamma$ resonance within generic two-Higgs-doublet models.