Not yet reviewed by Pith; the record is open.

Models of Cold Dark Matter predict that the distribution of dark matter in galaxy clusters should be cuspy, centrally concentrated. Constant density cores would be strong evidence for beyond-CDM physics, such as Self-Interacting Dark Matter (SIDM). An observable consequence would be oscillations of the Brightest Cluster Galaxy (BCG) in otherwise relaxed galaxy clusters. Offset BCGs have indeed been observed - but only interpreted via a simplified, analytic model of oscillations. We compare these observations to the BAHAMAS-SIDM suite of cosmological simulations, which include SIDM and a fully hydrodynamical treatment of star formation and feedback. We predict that the median offset of BCGs increases with the SIDM cross-section, cluster mass and the amount of stellar mass within 10kpc, while CDM exhibits no trend in mass. Interpolating between the simulated cross-sections, we find that the observations (of 10 clusters) is consistent with CDM at the ~1.5$\sigma$ level, and prefer cross-section $\sigma$/m < 0.12(0.39)cm$^2$/g at 68% (95%) confidence level. This is on the verge of ruling out velocity-independent dark matter self-interactions as the solution to discrepancies between the predicted and observed behaviour of dwarf galaxies, and will be improved by larger surveys by Euclid or SuperBIT.