The Fourier vanishing conjecture for hyper-Kähler varieties

Let $X$ and $Y$ be hyper-Kähler varieties of dimension $2n$, and let $\mathfrak{F}$ and $\mathfrak{F}^{-1}$ be the Fourier transforms and their inverses considered in the source, with graded components $\mathfrak{F}_i$ and $\mathfrak{F}^{-1}_j$. Fourier vanishing conjecture. For integers $0\leq i,j<2n$,

$$\mathfrak{F}_{2i+1}=0\in\operatorname{CH}^{2i+1}(Y\times X),\qquad \mathfrak{F}^{-1}_{2j+1}=0\in\operatorname{CH}^{2j+1}(X\times Y).$$

For integers $0\leq i,j\leq2n$ with $i+j\neq2n$,

$$\mathfrak{F}_{2i}\circ\mathfrak{F}^{-1}_{2j}=0\in\operatorname{CH}^{2i+2j-2n}(X\times X),$$ $$\mathfrak{F}^{-1}_{2i}\circ\mathfrak{F}_{2j}=0\in\operatorname{CH}^{2i+2j-2n}(Y\times Y).$$

This is part of the conjectural Fourier-transform package used to control the grading of motivic correspondences; its resolution status is not specified in the source.