  The communication complexity of achieving secret key (SK) capacity in the
multiterminal source model of Csisz$\'a$r and Narayan is the minimum rate of
public communication required to generate a maximal-rate SK. It is well known
that the minimum rate of communication for omniscience, denoted by
$R_{\text{CO}}$, is an upper bound on the communication complexity, denoted by
$R_{\text{SK}}$. A source model for which this upper bound is tight is called
$R_{\text{SK}}$-maximal. In this paper, we establish a sufficient condition for
$R_{\text{SK}}$-maximality within the class of pairwise independent network
(PIN) models defined on hypergraphs. This allows us to compute $R_{\text{SK}}$
exactly within the class of PIN models satisfying this condition. On the other
hand, we also provide a counterexample that shows that our condition does not
in general guarantee $R_{\text{SK}}$-maximality for sources beyond PIN models.
