Existence of a section after a degenerate twistor deformation of a Lagrangian fibration

Let $\pi\colon X\to B$ be a Lagrangian fibration with irreducible fibers, and suppose that the third Betti number of $X$ satisfies

$$b^3(X)=0.$$

Degenerate twistor deformation conjecture. There exists a degenerate twistor deformation of $\pi$ admitting a holomorphic section. This conjecture would remove the torsion-freeness assumption on $H^2(B,\Gamma)$ from the preceding theorem on sections in the Shafarevich–Tate family. It is presented as an open expectation in the source.