The converse invariant conjecture for complexified Farahat-Higman rings of wreath products

Let ${\Gamma}$ and $G$ be finite groups. For each $n$, consider the complexified rings $\mathbb C\otimes_{\mathbb Z}\mathcal G_{\Gamma}(n)$ and $\mathbb C\otimes_{\mathbb Z}\mathcal G_G(n)$. Converse invariant conjecture. If these rings are isomorphic for every $n$, then $|{\Gamma}^{re}_*|=|G^{re}_*|$, in addition to the identity $|{\Gamma}_*|=|G_*|$. This is the proposed converse to the preceding theorem, which shows that the complexified rings depend only on the numbers of conjugacy classes and real nonidentity conjugacy classes; the converse is left as a conjecture in the source.