Dimitrov's abelian-variety height lower-bound conjecture

Let $A$ be an abelian variety and let $\lambda:A\dashrightarrow\mathbb{P}^1$ be a rational function, both defined over a number field $K$. Let $h$ be the standard height on $\mathbb{P}^1$, and let $\widehat h$ be the canonical height associated with a fixed symmetric projective embedding of $A$. Dimitrov's conjecture. There is a positive lower bound

$$\inf_P\frac{1+h(\lambda(P))}{\widehat h(P)}>0,$$

where $P$ ranges over $K$-rational points in $A(K)\cap\operatorname{dom}(\lambda)$ that do not lie in a translate $Q+B$ of a positive-dimensional abelian subvariety contracted by $\lambda$, meaning $\lambda(Q+B)=\lambda(Q)$. In particular, when $A$ is simple and $\lambda$ is non-constant, the same assertion holds over all such $K$-rational points. This is the stronger second version of the proposed abelian-variety conjecture and remains open.