Boundedness conjecture for extremal classes on birational irreducible holomorphic symplectic manifolds

Let $X$ be a projective irreducible holomorphic symplectic manifold, let $\Lambda=H^{1,1}(X,\mathbb Z)$, and let $\operatorname{Nef}_Y^*$ denote the dual of the nef cone of a birational irreducible holomorphic symplectic manifold $Y$. An integral, primitive, and extremal class $e$ in $\operatorname{Nef}_Y^*$ is an integral primitive class spanning an extremal ray of this cone.

Boundedness conjecture. Let $Y$ be an irreducible holomorphic symplectic manifold birational to $X$. The set

$$\{(e,e): e \text{ is an integral, primitive, and extremal class in } \operatorname{Nef}_Y^*\}$$

is bounded below by a constant depending only on the birational class of $X$.

This is a boundedness assertion for extremal classes in the nef-dual, or Mori, cones of all irreducible holomorphic symplectic manifolds birational to a fixed $X$. It is part of the framework underlying the Kawamata–Morrison cone conjecture for varieties of K3$^{[n]}$ or generalized Kummer deformation type.