The cancellation conjecture for surjective self-morphisms of projective varieties

Let $X$ be a projective variety defined over a number field $K$, and let $f\colon X\longrightarrow X$ be a surjective self-morphism. Cancellation conjecture. There exists a non-negative integer $s_{0}$ such that, for all $x,y\in X(K)$, if

$$f^{s}(x)=f^{s}(y)$$

for some $s\geq 0$, then

$$f^{s_{0}}(x)=f^{s_{0}}(y).$$

The conjecture extends the cancellation theorem proved for projective curves and is motivated by the preimages question for arithmetic dynamical systems. Its validity in higher dimensions is not established by the supplied text.