The Planck collaboration has recently published precise and resolved measurements of the Sunyaev-Zeldovich effect in Abell 1656 (the Coma cluster of galaxies), so directly gauging the electron pressure profile in the intracluster plasma. On the other hand, such a quantity may be also derived from combining the density and temperature provided by X-ray observations of the thermal bremsstrahlung radiation emitted by the plasma. We find a model-independent tension between the SZ and the X-ray pressure, with the SZ one being definitely lower by 15-20%. We propose that such a challenging tension can be resolved in terms of an additional, non-thermal support to the gravitational equilibrium of the intracluster plasma. This can be straightforwardly included in our Supermodel, so as to fit in detail the Planck SZ profile while being consistent with the X-ray observables. Possible origins of the nonthermal component include cosmic-ray protons, ongoing turbulence, and relativistic electrons; given the existing observational constraints on the first two options, here we focus on the third. For this to be effective, we find that the electron population must include not only an energetic tail accelerated to gamma> 10^3 responsible for the Coma radiohalo, but also many more, lower energy electrons. The electron acceleration is to be started by merging events similar to those which provided the very high central entropy of the thermal intracluster plasma in Coma.
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