Căldăraru's conjecture on twisted K3 Fourier–Mukai partners

Let $(X,\alpha)$ be a twisted K3 surface, where $\alpha$ is a Brauer class, and let $X'$ be an untwisted Fourier–Mukai partner of $X$. For a Hodge isometry

$$g:T_{X'}\cong T_X,$$

consider the pullback Brauer class $g^*\alpha$ on $X'$. Căldăraru's conjecture. The twisted K3 surface $(X',g^*\alpha)$ is a Fourier–Mukai partner of $(X,\alpha)$ for every such $X'$ and every such Hodge isometry $g$. The conjecture is presented as equivalent to the surjectivity of the map described in the surrounding discussion, while injectivity is known; its resolution is not specified in the supplied text.