Hassett–Tschinkel's line-square conjecture for Lagrangian planes

Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type, meaning that it is deformation equivalent to a Hilbert scheme of $n$ points on a $K3$ surface. Let $\mathbb{P}^n\subset X$ be a smoothly embedded Lagrangian $n$-plane, and let $\ell\in H_2(X,\mathbb{Z})$ be the class of a line in $\mathbb{P}^n$. Hassett–Tschinkel's line-square conjecture. Then

$$(\ell,\ell)=-\frac{n+3}{2}.$$

This is the specialization to varieties of $K3^{[n]}$-type of the proposed description of extremal curve classes. The paper presents it as an expectation, and no resolution is supplied in the given text.