David–Philippon's height lower-bound conjecture for subvarieties of abelian varieties

Let $A$ be an abelian variety defined over a number field $k$, equipped with an ample symmetric line bundle $L$. Let $V$ be a proper $k$-irreducible subvariety of $A$ that is not a union of torsion subvarieties. Let $s$ be the dimension of the smallest algebraic subgroup containing $V$, and let $\widehat{h}_L(V)$ and $\deg_L(V)$ denote respectively the normalized height and the degree of $V$ with respect to $L$.

David–Philippon's conjecture. There is a constant $c(A,L)$ depending only on $A$ and $L$ such that

$$\frac{\widehat{h}_{L}(V)}{\deg_L(V)}\geq c(A,L)\deg_L(V)^{-\frac{1}{s-\dim V}}.$$

This is a lower-bound conjecture for the normalized heights of non-torsion subvarieties of abelian varieties, generalizing the problem posed by David and Philippon and relating to the Bogomolov and Lehmer problems. Its resolution is not specified in the source.