L-equivalence implies Fourier–Mukai partnership for cubic fourfolds

Let $Y$ and $Y'$ be cubic fourfolds. Write $\mathcal{A}_Y$ for the Kuznetsov component of $Y$, defined by

$$\mathcal{A}_Y:=\langle\mathcal{O}_Y,\mathcal{O}_Y(1),\mathcal{O}_Y(2)\rangle^\perp\subset D^b(\operatorname{Coh}(Y)).$$

The cubic fourfolds are FM partners if there is a Fourier–Mukai equivalence $\mathcal{A}_Y\simeq\mathcal{A}_{Y'}$. L-equivalence conjecture for cubic fourfolds. If $Y$ and $Y'$ are L-equivalent cubic fourfolds, then they are FM partners. This is known in the paper for very general cubic fourfolds in the specified Hassett divisors, but the unrestricted assertion for all cubic fourfolds is posed as an open question.