Stability of the symplectomorphism group of rational surfaces

We apply Zhang's almost K\"ahler Nakai-Moishezon theorem and Li-Zhang's comparison of $J$-symplectic cones to establish a stability result for the symplectomorphism group of a rational $4$-manifold $M$ with Euler number up to $12$. As a corollary, we also derive a stability result for the space of embedded symplectic balls in $M$. A noteworthy feature of our approach is that we systematically explore various spaces and groups associated to a symplectic cohomology class $u$ rather than with a single symplectic form $\omega$. To this end, we prove a weaker version of the tamed $J$-inflation procedures of D. McDuff and O. Buse that fixes a gap in their original formulations.