A conjectural bound on the second Betti number for hyper-K\"ahler manifolds

In previous work, we noted that the known cases of hyper-Kahler manifolds satisfy a natural condition on the LLV decomposition of the cohomology; informally, the Verbitsky component is the dominant representation in the LLV decomposition. Assuming this condition holds for all hyper-Kahler manifolds, we obtain an upper bound for the second Betti number in terms of the dimension.