How delicate are the $f(R)$ gravity models with a disappearing cosmological constant?

We consider stability of spherically symmetric solutions in $f(R)$ gravity model proposed by Starobinsky. We find that the model suffers from a severe fine-tuning problem when applied to compact objects like neutron stars. The problem can be remedied by introducing a cutoff on the mass of the scalar degree of freedom present in the model. A new mass scale associated with neutron stars density is then required for the stabilities of $f(R)$ gravity solutions inside relativistic stars.