The Fourier decomposition conjecture for zero-cycles on hyperkähler varieties

Let $F$ be a hyperkähler variety of dimension $2d$, let $L\in\mathrm{CH}^2(F\times F)$ be a canonical cycle representing the Beauville–Bogomolov class, and define $\mathcal{F}:=e^L$. Set

$$\mathrm{CH}^{2d}(F)_{2s}:=\{\sigma\in\mathrm{CH}^{2d}(F):\mathcal{F}(\sigma)\in\mathrm{CH}^{2s}(F)\}.$$

Fourier decomposition conjecture. There exists such an $L$ inducing a canonical splitting

$$\mathrm{CH}^{2d}(F)=\bigoplus_{s=0}^d\mathrm{CH}^{2d}(F)_{2s},$$

and

$$\mathrm{CH}^{2d}(F)_{2s}=\langle l^{d-s}\rangle\cdot(L_*\mathrm{CH}^{2d}(F))^{\cdot s},$$

while

$$\mathrm{CH}^{2d}(F)_{2s}\supseteq P(l,D_1,\ldots,D_r)\cdot(L_*\mathrm{CH}^{2d}(F))^{\cdot s}$$

for every degree $2d-2s$ weighted homogeneous polynomial $P$ in $l$ and divisors $D_i$. This would extend the Fourier decomposition known in special hyperkähler examples to zero-cycles on general hyperkähler varieties.