  $B_s \to \mu^+ \mu^- \gamma$, measured at high $q^2$ as a partially
reconstructed decay, can probe the origin of the existing discrepancies in
semi-leptonic $b \to s$ and $b \to c$ decays. We perform a complete study of
this possibility. We start by reassessing the alleged discrepancies, with a
focus on a unified EFT description. Using the SMEFT, we find that the tauonic
Wilson coefficient required by $R(D^{(*)})$ implies a universal muonic Wilson
coefficient of precisely the size required by semi-muonic BR data and,
separately, by semi-muonic angular analyses. We thus identify reference
scenarios. Importantly, $B_s \to \mu^+ \mu^- \gamma$ offers a strategy to
access them without being affected by the long-distance issues that hamper the
prediction of semi-leptonic $B$ decays at low $q^2$. After quantifying to the
best of our knowledge the $B_s \to \mu^+ \mu^- \gamma$ experimental over the
long haul, we infer the $B_s \to \mu^+ \mu^- \gamma$ sensitivity to the
couplings relevant to the anomalies. In the example of the real-$\delta
C_{9,10}$ scenario, we find significances below 3$\sigma$. Such figure is to be
compared with other single-observable sensitivities that one can expect from
e.g. BR and angular data, whether at low or high $q^2$, and not affected by
long-distance issues such as narrow resonances or intermediate charmed di-meson
rescattering.
