Finite-kernel specialization for Lang–Néron groups

Let $B$ be a smooth variety over a field $k$, let $K=k(B)$, and let $A$ be an abelian $K$-variety. Assume that $A$ has semistable reduction at every point of $B^{(1)}$. For any projective embedding $B\hookrightarrow\mathbf{P}^N$, there is a specialization homomorphism

$$\operatorname{LN}(A,K/k)\to \operatorname{LN}(A_s,E/k),$$

associated with a hyperplane section $h$ and its residue field $E=k(h)$. Finite-kernel specialization conjecture. There exists a smooth, geometrically connected hyperplane section $h$ of $B$ such that $A$ has good reduction at $h$ and the kernel of this specialization homomorphism is finite, with $E=k(h)$. This conjecture is introduced for use in a subsequent theorem; no proof or resolution is given in the supplied text.