Lang–Silverman conjecture for Zariski-dense endomorphism orbits

Let $g\geq 1$ be an integer. For any number field $K$, let $A/K$ be an abelian variety of dimension $g$, let $L$ be an ample symmetric line bundle on $A$, and let $P\in A(K)$. Write $\overline{\operatorname{End}(A)\cdot P}=A$ when the endomorphism orbit of $P$ is Zariski dense in $A$. The height $\widehat{h}_{A,L}$ is the Néron–Tate height associated to $L$, and $\mathop{h_{\mathrm{F}^+}}(A/K)$ is the relative Faltings height.

Lang–Silverman conjecture. There exists a positive quantity $c_{23}=c_{23}(K,g)$ such that

\overline{\operatorname{End}(A)\cdot P}=A \quad\Longrightarrow\quad \widehat{h}_{A,L}(P)\geq c_{23}\max\left\\{\mathop{h_{\mathrm{F}^+}}(A/K),1\right\\}.

This is the first version of the Lang–Silverman philosophy presented in the paper; the source notes that related formulations are known to be false in full generality, while the generic-point lower-bound principle remains the subject of the conjecture.