A specialisation theorem for Lang-N\'eron groups

We show that, for a polarised smooth projective variety $B \hookrightarrow \mathbb{P}^n_k$ of dimension $\geq 2$ over an infinite field $k$ and an abelian variety $A$ over the function field of $B$, there exists a dense Zariski open set of smooth geometrically connected hyperplane sections $h$ of $B$ such that $A$ has good reduction at $h$ and the specialisation homomorphism of Lang-N\'eron groups at $h$ is injective (up to a finite $p$-group in positive characteristic $p$). This gives a positive answer to a conjecture of the first author, which is used to deduce a negative definiteness result on his refined height pairing. This also sheds a new light on N\'eron's specialisation theorem.