L-equivalence conjecture for cubic fourfold Fourier–Mukai partners

Let $X$ and $Y$ be cubic fourfolds over the complex numbers. Their Kuznetsov components are the admissible subcategories $\mathcal{A}_X$ and $\mathcal{A}_Y$ in the semiorthogonal decompositions

$$\mathcal{D}^{b}(X)\simeq\langle\mathcal{A}_X,\mathcal{O}_X,\mathcal{O}_X(1),\mathcal{O}_X(2)\rangle.$$

Two cubic fourfolds are L-equivalent if their classes in the Grothendieck ring of varieties become equal after multiplication by a power of the class of the affine line. They are Fourier–Mukai partners when their Kuznetsov components are equivalent. L-equivalence conjecture for cubic fourfolds. If $X$ and $Y$ are L-equivalent, then they are Fourier–Mukai partners:

$$X\sim_{L}Y\implies\mathcal{A}_X\simeq\mathcal{A}_Y.$$

This is proposed as a cubic-fourfold instance of the conjectural implication from L-equivalence to derived equivalence for projective hyperkähler manifolds. The conjecture has also appeared recently in the cited literature and remains open.