Generalized Fermat's conjecture for compatible systems of endomorphisms

Let $K$ be an arithmetic function field equipped with a proper adelic structure having Northcott's property, let $X$ be a geometrically integral projective scheme over $K$, let $L$ be an ample line bundle on $X$, and let $F=\{f_N\}_{N=N_0}^{\infty}$ be a system of endomorphisms compatible with $L$. Let $h_F$ be its height function, let $Y$ be an equidimensional subscheme of $X$, and set $Y_N=f_N^{-1}(Y)$. We say that $Y_N$ has Fermat's property over $K$ when

$$Y_N(K)\subseteq\{x\in X(K)\mid h_F(x)=0\}.$$

Generalized Fermat's conjecture. If there is $N_1\geqslant N_0$ such that $Y_N(K)$ is finite for every integer $N\geqslant N_1$, then there exists $N_2\geqslant N_0$ such that $Y_N$ has Fermat's property over $K$ for every $N\geqslant N_2$.

This proposes that eventual finiteness of the rational points on the inverse images forces those points to have canonical height zero. The paper presents it as a conjecture and gives evidence; the supplied material does not establish a general resolution.