Markman–Mukai lattice conjecture for twisted derived categories of hyper-Kähler varieties

Let $X$ and $Y$ be hyper-Kähler varieties of $K3^{[n]}$-type. Let $L(X)$ and $L(Y)$ be their Markman–Mukai lattices, with primitive generators $v$ and $w$ of the rank-one orthogonal complements $H^2(X)^\perp$ and $H^2(Y)^\perp$, respectively; these choices determine orientations $(L(X),v)$ and $(L(Y),w)$. The associated canonical classes are $\theta_v\in H^2(X,\mu_{2n-2})$ and $\theta_w\in H^2(Y,\mu_{2n-2})$. Markman–Mukai lattice conjecture. If there exists a Hodge isometry between the oriented Markman–Mukai lattices

$$\phi:(L(X),v)\longrightarrow (L(Y),w),$$

then there is an equivalence of bounded derived categories

$$D^b(X,[n\theta_v])\cong D^b(Y,[\epsilon n\theta_w]),$$

where $\epsilon=1$ if $\phi$ is orientation-preserving and $\epsilon=-1$ if $\phi$ is orientation-reversing. More broadly, the authors conjecture that a natural twisted derived category of every hyper-Kähler variety of $K3^{[n]}$-type is controlled by its Markman–Mukai lattice. The conjecture proposes a lattice-theoretic classification of these twisted derived categories, but the supplied text gives no resolution.