Huybrechts's Fourier–Mukai partners conjecture for cubic fourfolds

Let $X$ and $X'$ be smooth cubic fourfolds, and let $\mathcal{A}_X$ and $\mathcal{A}_{X'}$ denote their Kuznetsov components. Huybrechts's conjecture. If

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

then $X$ and $X'$ are birationally equivalent. Equivalently, Fourier–Mukai partnership implies birational equivalence. This conjecture is known in some special cases, including a very general cubic in $\mathcal{C}_{20}$, but remains open in general.