Polynomial canonical-height bound for integral points on abelian varieties

Let $B$ be a smooth projective curve, let $A/K$ be an abelian variety with a model $f\colon\mathcal{A}\to B$, where $\mathcal{A}_K=A$. Let $D\subset A$ be a reduced effective ample divisor, and let $\mathcal{D}$ be its Zariski closure in $\mathcal{A}$. For a symmetric ample line bundle $L$ on $A$, denote the canonical height by $\widehat{h}_{A,L}$. Polynomial height-bound conjecture. There exist $m,N>0$, depending only on $\mathcal{A},B,\mathcal{D}$, such that for every $(S,\mathcal{D})$-integral point $P\in A(K)$ with $S\subset B$, one has

$$\widehat{h}_{A,L}(P)\leq m(\#S+1)^N.$$

The proposed bound would yield polynomial counting estimates for integral points modulo the $K/\mathbb{C}$-trace. The text explains that the expected exponent is $N=1$ in two special cases and $N=2$ in general, while no comparable result was known for a general abelian variety.