Huybrechts' Hodge-theoretic characterization of Fourier–Mukai partners

Let $X$ and $Y$ be smooth cubic fourfolds, and write $\widetilde{\mathrm{H}}(X)$ and $\widetilde{\mathrm{H}}(Y)$ for the Addington–Thomas Hodge structures associated with their Kuznetsov components. A Hodge isometry is an isometry preserving the Hodge structures, and an orientation-preserving Hodge isometry also preserves the natural orientations of the negative directions. Huybrechts' conjecture. The cubic fourfolds $X$ and $Y$ are Fourier–Mukai partners if and only if there exists an orientation-preserving Hodge isometry

$$\widetilde{\mathrm{H}}(X)\simeq\widetilde{\mathrm{H}}(Y).$$

This would characterize Fourier–Mukai equivalence purely through the associated Hodge lattices; the source gives no resolution status.