Căldăraru's twisted derived equivalence conjecture for K3 surfaces

Let $X$) and $X'$ be algebraic K3 surfaces, and let $\alpha\in H^2(X,\mathcal O_X^*)$ and $\alpha'\in H^2(X',\mathcal O_{X'}^*)$ be torsion classes. Define the twisted transcendental lattices by lifting these classes to $H^2(X,\mathbb Q)$ and $H^2(X',\mathbb Q)$ and setting

$$T(X,\alpha):=\operatorname{Ker}(\alpha:T(X)\to\mathbb Q/\mathbb Z),$$

and similarly for $T(X',\alpha')$.

Căldăraru's conjecture. There exists an exact equivalence between the derived categories of twisted coherent sheaves

$$D^{\operatorname b}(\operatorname{Coh}_\alpha(X))\simeq D^{\operatorname b}(\operatorname{Coh}_{\alpha'}(X'))$$

if and only if there exists a Hodge isometry

$$T(X,\alpha)\cong T(X',\alpha').$$

This conjecture proposes a lattice-theoretic characterization of equivalence between derived categories of twisted coherent sheaves on algebraic K3 surfaces. The source notes that the analytic analogue had recently been established, while the algebraic statement was proposed by Căldăraru.