  We compute the decomposition numbers of the unipotent characters lying in the
principal $\ell$-block of a finite group of Lie type $B_{2n}(q)$ or $C_{2n}(q)$
when $q$ is an odd prime power and $\ell$ is an odd prime number such that the
order of $q$ mod $\ell$ is $2n$. Along the way, we extend to these finite
groups the results of \cite{DVV19} on the branching graph for Harish-Chandra
induction and restriction.
