Zhang's Zariski dense orbit conjecture for polarized endomorphisms

Let ${\mathbf{k}}$ be an algebraically closed field of characteristic $0$. Let $X$ be an irreducible projective variety over ${\mathbf{k}}$, and let $f:X\to X$ be a polarized endomorphism, meaning that there are an ample line bundle $L$ on $X$ and an integer $d>1$ such that $f^*L=L^{\otimes d}$. For $x\in X({\mathbf{k}})$, write

$$O_f(x):=\{f^n(x)\mid n\geq 0\}.$$

Zhang's conjecture. There exists a point $x\in X({\mathbf{k}})$ whose orbit $O_f(x)$ is Zariski dense in $X$. This is a polarized special case of the Zariski dense orbit problem and is implied by the broader conjecture above; it remains open in general.