Equivalences via twisted hyperholomorphic sheaves from transverse Lagrangian fibrations

Following ideas of Kapustka-Kapustka, we use Lagrangian fibrations to construct twisted hyperholomorphic sheaves on products of hyperkähler manifolds of K3$^{[n]}$- and OG10-type. As applications, we prove the Lefschetz standard conjecture and the D-equivalence conjecture for hyperkähler manifolds of OG10-type.