A Reduction theorem for the $W$-graph decomposition conjecture

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 $W_2$.