  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.
