  In \cite{CDD22} we investigated the structure of $\ast$-isomorphisms between
von Neumann algebras $L(\Gamma)$ associated with graph product groups $\Gamma$
of flower-shaped graphs and property (T) wreath-like product vertex groups as
in \cite{CIOS21}. In this follow-up we continue the structural study of these
algebras by establishing that these graph product groups $\Gamma$ are entirely
recognizable from the category of all von Neumann algebras arising from an
arbitrary non-trivial graph product group with infinite vertex groups. A
sharper $C^*$-algebraic version of this statement is also obtained. In the
process of proving these results we also extend the main $W^*$-superrigidity
result from \cite{CIOS21} to direct products of property (T) wreath-like
product groups.
