Fix a prime number $l$. Let $K$ be a finite extension of $\mathbb{Q}$ and $R$ be the ring of integers in $K$.
In Chapter 2 of the Storrs volume (Cornell-Silverman) it is claimed that it is not necessarily true that for a generically proper semiabelian variety $A$ over $R$ the height of $A/A[l^n]$ (for any positive integer $n$) is equal to the height of $A$ (but it is true that the former eventually stabilizes). Do you have an explicit counterexample? Is there a counterexample where the stable value is different from the height of $A$? Given an arbitrary positive integer $n_0$, can you give an example where the value stabilizes exactly at $n_0$?