Fun idea. I never heard of Pion stars before :-)
A star could be defined as massive body that counterbalances the effect of the gravitational force by a source of energy (in standard stars, nuclear fusion). That equilibrium between gravity and radiative pressure guarantees the star's stability.
In pion stars, what would be the source of energy?
Without one, a gravitational induced pion-condensate, would immediately collapse into a black hole, especially if its mass is as huge as 250 solar masses. Beyond the black hole horizon, I can then this object evolving happily as a an hypothetical quark star, or maybe even a strange star (plasma of u, d and s quarks).
This is a complicated question to answer and the answer is in the paper. If I have well understood, the entire system relies on chiral perturbation theory and the bounding of the system is connected to the isospin symmetry. The isospin density is connected to the charge density that counterbalanced the gravitational pressure.
Bosonic stars are actually good candidate to mimic black holes. Their radius is however much larger.
I had a quick look at the paper. I am not too sure I got an answer to my question, because I honestly understood less than half of it.
I believe what they call the pionic pressure would be what counterbalances gravity. Then, I have to extrapolate to what I know. Without entering the details, would it be that the compression by gravity of the pions force-stacks each of them into a state resulting in a counter pressure, analogously to the state degeneracy of electrons or neutrons that stabilizes white dwarfs and neutron stars respectively?
In the end, there is no need for a source of energy. Quantum physics rules do the job.
In terms of gut feelings (i.e. with no logic), I am quite doubtful of the past or present existence of these pion-electron stars, and that even if I love exotic things ;-)
In short, the weak isospin interactions of the pions balance the gravitational collapse. It is sufficient and you don't need to invoke anything new. (On top of that, they have verified that the pion condensate is stable, as pion are unstable particles.)