Symplectic mapping class groups of blowups of tori

Let $\omega$ be a Kaehler form on the real $4$-torus $T^4$. Suppose that $\omega$ satisfies an irrationality condition which can be achieved by an arbitrarily small perturbation of $\omega$. This note shows that the smoothly trivial symplectic mapping class group of the one-point symplectic blowup of $(T^4,\omega)$ is infinitely generated.