The rational Lagrangian fibration conjecture for irreducible symplectic varieties

Let $X$ be an irreducible symplectic variety, let $0\neq L\in \operatorname{Pic}(X)\cap \operatorname{\overline{\mathscr{B}\mathscr{K}}}_X$ be a primitive line bundle, and suppose that $q(L)=0$. Here $\operatorname{\overline{\mathscr{B}\mathscr{K}}}_X$ denotes the closure of the birational Kähler cone, and $|L|$ denotes the complete linear system of $L$.

Rational Lagrangian fibration conjecture. One has

$$\dim h^0(X,L)=n+1,$$

and $|L|$ induces a birational Lagrangian fibration to $\mathbb{P}^n$.

This conjecture predicts both the number of sections of a primitive isotropic line bundle in the birational Kähler cone and the existence of the associated birational Lagrangian fibration. Its status is not specified in the supplied source context.