  The decay $B^0_s\to\mu^+\mu^-$ is a key probe for the search of physics
beyond the Standard Model. While the current measurements of the corresponding
branching ratio agree with the Standard Model within the uncertainties,
significant New-Physics effects may still be hiding in $B^0_s\to\mu^+\mu^-$. In
order to reveal them, the observable $\mathcal{A}^{\mu\mu}_{\Delta \Gamma_s}$,
which is provided by the decay width difference $\Delta\Gamma_s$ of the
$B^0_s$-meson system, plays a central role. We point out that a measurement of
a CP-violating observable ${\cal S}_{\rm \mu\mu}$, which is induced through
interference between $B^0_s$-$\bar B^0_s$ mixing and $B_s\to\mu^+\mu^-$ decay
processes, is essential to obtain the full picture, in particular to establish
new scalar contributions and CP-violating phases. We illustrate these findings
with future scenarios for the upgrade(s) of the LHC, exploiting also relations
which emerge within an effective field theory description of the Standard
Model, complemented with New Physics entering significantly beyond the
electroweak scale.
