Hassett–Tschinkel's effective-cone conjecture for irreducible holomorphic symplectic varieties

Let $X$ be an irreducible holomorphic symplectic variety with polarization $H$, and let $N_1(X,\mathbb{Z})$ be its group of curve classes modulo homological equivalence. Write $\operatorname{NE}_1(X)$ for the cone of effective curves, and let $(\cdot,\cdot)$ be the Beauville–Bogomolov form. Hassett–Tschinkel's conjecture. There is a positive rational constant $c_X$, depending only on the deformation class of $X$, such that

$$\operatorname{NE}_1(X)=\langle C\in N_1(X,\mathbb{Z})\mid H\cdot C>0\ \mathrm{and}\ (C,C)\geq -c_X\rangle.$$

Moreover, if $X$ contains a smoothly embedded Lagrangian $n$-plane $\mathbb{P}^n\subset X$ and $\ell\in\operatorname{NE}_1(X)$ is the class of a line in $\mathbb{P}^n$, then

$$(\ell,\ell)=-c_X.$$

The paper later refers to this as disproven, so the conjecture is not valid in its stated generality.