Voisin map action conjecture on zero-cycles of higher Fano varieties of planes

Let $Y$ be a cubic hypersurface and let $X=F_r(Y)$ be the variety of $r$-planes in $Y$, equipped with the Voisin rational self-map $\Psi:X\dashrightarrow X$. For $r\geq 2$ and $z\in CH_0(X)_{\mathrm{hom}}$, Voisin map action conjecture.

$$\Psi_*z=(-2)^{r+1}z.$$

The paper derives this prediction from the generalized Bloch conjecture and proves it when $r=2$ for a general cubic $8$-fold; the assertion remains conjectural for general $r\geq2$.