The fixed-locus constant-cycle conjecture for Voisin maps

Let $X=F_r(Y)$ with $r\geq2$, let $\Psi:X\dashrightarrow X$ be the Voisin map, and let $F\subset X$ be the closure of its fixed locus. Fixed-locus constant-cycle conjecture. The variety $F$ is a constant-cycle subvariety for $r\geq2$. The source says this follows immediately from the preceding Voisin-map action conjecture; no independent resolution is supplied.