The Fermat conjecture inside projective varieties

Let $V\subseteq\mathbb P^m$ be a nonempty closed subscheme over $\mathbb Q$. Say that Fermat holds inside $V$ if, for every $a\in\mathbb Z^{m+1}$ with $a_i\neq 0$, there is $n_0\in\mathbb N$ such that the rational points of

$$X_n\subseteq V: a_0x_0^n+\dotsb+a_mx_m^n=0$$

are trivial for all $n\geq n_0$. Define $Z\subseteq\mathbb P^m$ by

$$Z=\bigcap_j\bigcup_{i\neq j}V_+(x_i^2x_j-x_j^3).$$

Fermat conjecture. Fermat holds inside $V$ if and only if $(V\cap Z)(\mathbb Q)$ is trivial. This gives a proposed characterization of the projective varieties for which the generalized Fermat problem is solvable; the conjecture is presented as open in the paper.