Can AGN feedback break the self-similarity of galaxies, groups, and clusters?

It is commonly thought that AGN feedback can break the self-similar scaling relations of galaxies, groups, and clusters. Using high-resolution 3D hydrodynamic simulations, we isolate the impact of AGN feedback on the $L_{\rm x}-T_{\rm x} $ relation, testing the two archetypal and common regimes, self-regulated mechanical feedback and a quasar thermal blast. We find that AGN feedback has severe difficulty in breaking the relation in a consistent way. The similarity breaking is directly linked to the gas evacuation within $R_{500}$, while the central cooling times are inversely proportional to the core density. Breaking self-similarity implies thus breaking the cool core, morphing all systems to non-cool-core objects, which is in clear contradiction with the observed data populated by several cool-core systems. Self-regulated feedback, which quenches cooling flows and preserves cool cores, prevents the dramatic evacuation and similarity breaking at any scale; the relation scatter is also limited. The impulsive thermal blast can break the core-included $L_{\rm x}-T_{\rm x}$ at $T_{500} < 1$ keV, but substantially empties and overheats the halo, generating a perennial non-cool-core group, as experienced by cosmological simulations. Even with partial evacuation, massive systems remain overheated. We show the action of purely AGN feedback is to lower the luminosity and heating the gas, perpendicular to the fit.