  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.
