Conjecture on bijective actions on dense sets of rational points

Let $X$ be a proper algebraic variety over a finitely generated field $K$ of characteristic zero, and let $f:X\to X$ be a morphism. Suppose there is a subset $S\subseteq X(K)$ that is Zariski dense in $X$ and on which $f$ induces a bijection. Dense-set bijection conjecture. Then $f$ is an automorphism. This conjecture asks whether bijectivity on a Zariski-dense set of rational points forces a self-morphism of a proper variety to be invertible, extending the analogy with finiteness and Mordell–Weil phenomena for abelian varieties over finitely generated fields.