  We consider the $B^0_s \to \mu^{+} \mu^{-} \gamma$ effective lifetime, and
the related CP-phase sensitive quantity $A_{\Delta\Gamma_s}^{\mu \mu \gamma}$,
as a way to obtain qualitatively new insights on the current $B$-decay
discrepancies. Through a fit comparing pre- to post-Moriond-2021 data we
identify a few theory benchmark scenarios addressing these discrepancies, and
featuring large CP violation in addition. We then explore the possibility of
telling apart these scenarios with $A_{\Delta\Gamma_s}^{\mu \mu \gamma}$, once
resonance-modeling and form-factor uncertainties are taken into account. We do
so in both regions of low and high invariant di-lepton mass-squared $q^2$. For
low $q^2$, we show how to shape the integration range in order to reduce the
impact of the $\phi$-resonance modelling on the $A_{\Delta\Gamma_s}^{\mu \mu
\gamma}$ prediction. For high $q^2$, we find that the corresponding pollution
from broad-charmonium resonances has a surprisingly small effect on
$A_{\Delta\Gamma_s}^{\mu \mu \gamma}$. This is due to a number of
cancellations, that can be traced back to the complete dominance of
semi-leptonic operator contributions for high $q^2$ -- at variance with low
$q^2$ -- and to $A_{\Delta\Gamma_s}^{\mu \mu \gamma}$ behaving like a
ratio-of-amplitudes observable. Our study suggests that
$A_{\Delta\Gamma_s}^{\mu \mu \gamma}$ is -- especially at high $q^2$ -- a
potentially valuable probe of short-distance CP-violating effects in the very
same Wilson coefficients that are associated to current $b \to s$
discrepancies. Its discriminating power, however, relies on progress in
form-factor uncertainties. Interestingly, high $q^2$ is the region where $B^0_s
\to \mu^{+} \mu^{-} \gamma$ is already being accessed experimentally, and the
region where form factors are more accessible through non-perturbative QCD
methods.
