Symplectic isotopy conjecture for ruled surfaces

Let $M_g=\Sigma_g\times S^2$ be a minimal topologically trivial ruled surface equipped with a symplectic form $\omega$. A symplectomorphism of $(M_g,\omega)$ is symplectically isotopic to the identity if it lies in the identity component of the symplectomorphism group, and is smoothly isotopic to the identity if it lies in the identity component of the diffeomorphism group. Symplectic isotopy conjecture. A symplectomorphism of $(M_g,\omega)$ is symplectically isotopic to the identity if and only if it is smoothly isotopic to the identity. This was proposed as an open problem for minimal ruled surfaces; the source does not report a resolution, although it records related stability results for ruled surfaces and their blow-ups.