Finiteness conjecture for compactified torsors of Lagrangian fibrations

Let $\pi_0:X^0\rightarrow\mathbb{P}^n$ be a fixed Lagrangian fibration. A compactified torsor over $X^0$ is a Lagrangian fibration $\pi:X\rightarrow\mathbb{P}^n$ such that, over the complement of the discriminant locus, $X$ is a torsor over $X^0$.

Finiteness conjecture for compactified torsors. Up to deformation, there are finitely many compactified torsors $\pi:X\rightarrow\mathbb{P}^n$ associated to the fixed Lagrangian fibration $\pi_0:X^0\rightarrow\mathbb{P}^n$.

The assertion asks for finiteness of the possible compactifications of torsors over a given Lagrangian fibration. The source presents it as a conjectural finiteness statement; no resolution is supplied here.