The derived Torelli conjecture for K3 categories of cubic fourfolds

Let $X,X'\subset\mathbb{P}^5$ be smooth cubic fourfolds, let $\mathcal{A}_X$ and $\mathcal{A}_{X'}$ be their K3 categories, and let $\widetilde H(X,\mathbb{Z})$ and $\widetilde H(X',\mathbb{Z})$ be their extended Hodge structures with the relevant orientation. Derived Torelli conjecture. There exists an exact linear equivalence

$$\mathcal{A}_X\simeq\mathcal{A}_{X'}$$

if and only if there exists an orientation-preserving Hodge isometry

$$\widetilde H(X,\mathbb{Z})\simeq\widetilde H(X',\mathbb{Z}).$$

The source places this among the results needed to establish the full K3-surface-like picture and explicitly says that the corresponding statement is not known in the required generality.