The unbounded-degree ruled-divisors conjecture for projective IHS manifolds

Let $X$ be a projective irreducible holomorphic symplectic manifold, and let a polarization on $X$ determine degrees of curves. The unbounded-degree ruled-divisors conjecture. $X$ contains infinitely many integral rational curves $C$ whose degrees with respect to a polarization are unbounded and each of which rules an ample divisor. This is a weaker form of the preceding conjecture, introduced as the statement studied in the paper. The paper proves related existence results for projective IHS manifolds of $K3^{[n]}$ or generalized Kummer type outside the cases excluded in its abstract, but the conjecture itself remains open in general.