Multi-indexed generalized Fermat's conjecture

Let $I_0=(N_1,\ldots,N_n)$, let $I,J\in(\mathbb{Z}_{\geqslant1})^n$, and write $I\geqslant I_0$ coordinatewise. Let $F$ be the multi-indexed system of endomorphisms with $f_I=f_{1,I(1)}\times\cdots\times f_{n,I(n)}$, let $Y$ be an equidimensional subscheme of $X$, and set $Y_I=f_I^{-1}(Y)$. Let $h_F$ be the associated height function. We say that $Y_I$ has Fermat's property over $K$ when

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

Multi-indexed generalized Fermat's conjecture. If there is $I_1\geqslant I_0$ such that $Y_I(K)$ is finite for every $I\geqslant I_1$, then there exists $I_2\geqslant I_0$ such that $Y_I$ has Fermat's property over $K$ for every $I\geqslant I_2$.

This is the multi-indexed analogue of the preceding generalized Fermat problem. The supplied material includes a multiplicative special case proved by a corollary, but does not resolve the stated general question.