Realization conjecture for almost toric fibrations and symplectic log Calabi–Yau divisors

Let $(X,c\omega)$ be a symplectic $4$-manifold. An almost toric fibration is a Lagrangian fibration with only nodal and elliptic singularities. Let $\mathcal{ATF}(X,\omega)$ denote the space of almost toric fibrations and let $\mathcal{LCY}(X,\omega)$ denote the space of symplectic log Calabi–Yau divisors; the map $\Phi$ sends an almost toric fibration to the symplectic log Calabi–Yau divisor given by the preimage of the boundary of its base. Realization conjecture. The map

$$\Phi:\mathcal{ATF}(X,\omega)\to\mathcal{LCY}(X,\omega)$$

is surjective. An affirmative answer would imply that every symplectic rational manifold with $\omega\cdot c_1>0$ admits an almost toric fibration. The conjecture is proved for $S^2\times S^2$ and $\mathbb{CP}^2\# l\overline{\mathbb{CP}}^2$ with $l\leq 3$ and arbitrary symplectic forms, but remains open in general.