Zhang's moduli-space conjecture for hyperkähler varieties of K3^[n]-type

Let $X$ be a hyperkähler variety of K3$^{[n]}$-type with $n\geq 2$. A K3 category is a category of K3 type, and let $\mathcal A_X^{[n]}$ denote its Hilbert-scheme-type construction. Zhang's conjecture. There exists a K3 category $\mathcal A_X$ such that $X$ is a moduli space of stable objects on $\mathcal A_X$, together with a canonical Brauer class $\theta_X\in\operatorname{Br}(X)$ and an exact linear equivalence

$$D^b(X,\theta_X^{\pm n})\simeq\mathcal A_X^{[n]}.$$

The conjecture predicts that every hyperkähler variety of K3$^{[n]}$-type admits a moduli-theoretic description through a K3 category, with its twisted derived category governed by the corresponding categorical Hilbert scheme. The stated corollary provides evidence in the case of Lagrangian fibered hyperkähler manifolds under the given hypotheses; the general assertion remains open.