The linear-subspace conjecture for inverse-periodic subvarieties of projective space

Let $X:=\mathbb{P}^n$, let $f:X\to X$ be an endomorphism with $\deg(f)\geq 2$, and let $Y$ be a closed subvariety of $X$. The linear-subspace conjecture. If $Y$ is $f^{-1}$-invariant or $f^{-1}$-periodic, then $Y$ is a linear subspace of $X$. This conjecture arises in the study of dynamical systems. The source lists known cases due to several works, but does not state that the conjecture is completely resolved.