  This article is part of a series with the aim of classifying all
non-hyperoctahedral categories of two-colored partitions. Those constitute by
some Tannaka-Krein type result the co-representation categories of a specific
class of quantum groups. However, our series of articles is purely
combinatorial. In Part I we introduced a class of parameters which gave rise to
many new non-hyperoctahedral categories of partitions. In the present article
we show that this class actually contains all possible parameter values of all
non-hyperoctahedral categories of partitions. This is an important step towards
the classification of all non-hyperoctahedral categories.
