The connectivity conjecture for toric fibrations

Let $(M,\omega)$ be a symplectic four-manifold equipped with toric fibrations, and let an almost toric fibration mean a fibration of $(M,\omega)$ obtained by allowing the nodal singularities and branch moves used in the preceding construction. Connectivity conjecture. Any two toric fibrations of $(M,\omega)$ can be connected by a path of almost toric fibrations of $(M,\omega)$, with $\omega$ fixed throughout this path. This would provide a geometric explanation for the equivalence of the different toric descriptions and their induced symplectic forms; the supplied text does not indicate whether the assertion has been proved or remains open.