The general-position conjecture for critically transverse automorphism orbits

Let $\Bbbk$ be an algebraically closed field of characteristic 0, let $X/\Bbbk$ be a projective variety, let $\sigma \in \operatorname{Aut}_{\Bbbk} X$, and let $Z \subseteq X$ be a closed subvariety. Here, $\sigma$ and $Z$ are in general position when $Z$ is homologically transverse to all $\sigma$-fixed subschemes of $X$. General-position conjecture. The collection $\{\sigma^n Z\}$ is critically transverse if and only if $\sigma$ and $Z$ are in general position. This is suggested by the preceding characterization of critical transversality for an algebraic-group action; the conjecture specializes to the case of a point proved in the cited work, while the stated projective-variety formulation remains unresolved here.