  We investigate the sensitivity of future proton-proton colliders to a contact
interaction of the form $1/\Lambda^2 (\bar b_L \gamma_\mu s_L)(\bar \mu_L
\gamma^\mu \mu_L)$ as indicated by the long-standing rare $B$-decay anomalies.
We include NLO QCD and electroweak effects and employ an optimized binning
scheme, and carefully validate our background calculation against ATLAS and CMS
data. We find that the FCC-hh with $40$ ab$^{-1}$ of luminosity is able to
exclude scales $\Lambda$ up to 26 TeV at $95 \%$ CL, and discover $\Lambda$ up
to 20 TeV. While this is not quite enough to exclude or discover the current
best-fit value of $39$ TeV, this can in principle be achieved with more
luminosity and/or higher energy, as we study quantitatively. Our analysis is
conservative in that it assumes only a $\bar b s \mu \mu $ contact interaction.
