The special Lagrangian fibration conjecture for rational elliptic surfaces

Let $X$ be a rational elliptic surface obtained by blowing up $\mathbb{CP}^2$ at the nine base points of a pencil of cubics, and let $\hat D$ be a smooth elliptic fiber. Rational elliptic surface fibration conjecture. The complement $X\setminus\hat D$ carries a special Lagrangian torus fibration over a disc with, generically, twelve nodal singular fibers; the monodromy of the affine structure around each singularity is conjugate to

$$\begin{pmatrix}1&1\\0&1\end{pmatrix},$$

and the monodromy along $\partial\hat B$ is trivial. This is expected from the projective-plane fibration by inserting one nodal fiber for each blow-up, but the source says that the required deformation theory was not yet established in general.