The period-index problem for hyperkähler manifolds

We conjecture that every unramified Brauer class $\alpha \in \text{Br}(X)$ on a projective hyperkähler manifold $X$ satisfies $\text{ind}(\alpha )\mid \text{per}(\alpha {)}^{\dim (X)\mathrm{/}2}$. We provide evidence for this conjecture by proving it for two large classes of projective hyperkähler manifolds: For projective hyperkähler manifolds admitting a Lagrangian fibration and for Hilbert schemes of K3 surfaces.