  Let $W$ be a finite Coxeter group and $\Omega$ be its $W$-graph algebra as
defined by Gyoja. The author's previous paper \cite{hahn2016wgraphs} considered
this algebra in some detail, proposed, and proved in some small cases the
$W$-graph decomposition conjecture. The purpose of the current paper is to
prove a reduction theorem for (a slightly stronger version of) that conjecture
to indecomposable Coxeter groups in the sense that the conjecture is true for
$W=W_1\times W_2$ if it holds for $W_1$ and $W_2$.
