Linearity conjecture for totally invariant prime divisors of projective space

Let $D\subset \mathbb{P}^{n}$ be a prime divisor. It is totally invariant if there exists a non-isomorphic endomorphism $f\colon \mathbb{P}^{n}\to\mathbb{P}^{n}$ such that

$$f^{-1}(D)=D.$$

Linearity conjecture. Every totally invariant prime divisor of $\mathbb{P}^{n}$ is linear.

This conjecture predicts that totally invariant divisors of projective space are unions of hyperplanes; for a prime divisor, this means a hyperplane. The paper establishes an upper bound for the degree of a totally invariant divisor and proves linearity when the divisor has isolated singularities, but the general statement remains open.