Constant-cycle fixed locus conjecture for the Voisin map

Let $X=F_r(Y)$ be as above, with Voisin rational self-map $\Psi:X\dashrightarrow X$, and let $F\subset X$ be the closure of its fixed locus. A closed subvariety is constant-cycle when all of its points are rationally equivalent in the ambient variety. Constant-cycle fixed-locus conjecture. The variety $F$ is a constant-cycle subvariety for $r\geq2$. The paper says this follows immediately from the conjectural action of $\Psi$ on homologically trivial zero-cycles; because that action is proved only in the case $r=2$, the general assertion remains open.