The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles

We show that birational hyper-K\"ahler varieties of $K3^{[n]}$-type are twisted derived equivalent with respect to some Brauer class. Furthermore, if a $K3^{[n]}$-type variety X admits a divisor class of divisibility 1 whose norm satisfies a congruence condition modulo 4, we show that any hyper-K\"ahler variety birational to X is derived equivalent to X. This verifies new cases of the D-equivalence conjecture in higher dimension. The Fourier-Mukai kernels of our (twisted) derived equivalences are constructed from Markman's projectively hyperholomorphic bundles.