Generic connectedness conjecture for symplectomorphism groups of rational surfaces

Let $X=\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$, and let $\omega$ be a generic symplectic form on $X$. Generic connectedness conjecture. The symplectomorphism group $\operatorname{Symp}(X,\omega)$ is connected. The conjecture is a weaker consequence of the generation conjecture because the existence of Lagrangian spheres is constrained by countably many numerical conditions on $[\omega]$. It is stated as open in the source.