The numerical Riemann–Roch polynomial conjecture for hyper-Kähler manifolds

Let $X$ be a hyper-Kähler manifold of dimension $2n$ with classes ${\mathsf l},{\mathsf m}\in H^2(X,{\bf Z})$ such that

$$\int_X{\mathsf l}^{2n}=0\quad\textnormal{and}\quad\int_X{\mathsf l}^n{\mathsf m}^n=n!.$$

Numerical Riemann–Roch conjecture. The Huybrechts–Riemann–Roch polynomial of $X$ satisfies

$$\forall k\in{\bf Z}\qquad P_{RR,X}(2k)=\chi({\bf P}^n,\mathscr{O}_{{\bf P}^n}(k+1))=\binom{k+1+n}{n}.$$

The assertion is motivated by the case where the isotropic class comes from a globally generated nef line bundle inducing a Lagrangian fibration. The paper presents it as open in the general numerical setting.