Projective-space quotient conjecture for bases of Lagrangian fibrations

Let $X$ be a $2n$-dimensional projective irreducible symplectic variety equipped with a Lagrangian fibration

$$X\to B.$$

Lagrangian-fibration base conjecture. The base $B$ is isomorphic to a quotient of

$$\mathbb{P}^n.$$

This conjecture extends the expected description of bases of Lagrangian fibrations on smooth irreducible symplectic manifolds to possibly singular irreducible symplectic varieties. The smooth case is known in dimension at most four, while the general assertion remains open.