Ample canonical height zero-set conjecture

Let $X$ be a smooth projective variety over $\overline{\mathbb Q}$, let $f$ be an endomorphism of $X$ with $\delta_f>1$, and let $K$ be a number field. Write $Z_f(K)$ for the set of $K$-rational points of $X$ at which the lower ample canonical height $\underline h_f$ vanishes. Ample canonical height zero-set conjecture. There is an $f$-invariant proper closed subset $V$ of $X$ such that $Z_f(K)\subseteq V$. This is proposed as a dynamical analogue of the Northcott finiteness theorem; the statement is known in several cases discussed in the paper, but is not clear in general. The assumption $\delta_f>1$ is necessary.