Elementary-divisor conjecture for critical groups of induced representations

Let $A$ be an abelian group of order $r$, let $\mathfrak{S}$ denote the tower of symmetric groups, and consider the differential tower of groups $A\wr\mathfrak{S}$. For $k\leq n$, let $V(U^kD^k)_n$ be the induced representation corresponding to $U^kD^k$, and let $p_j$ denote the rank sizes. Write $e_i$ for the non-unit elementary divisors of the critical group $K(V(D^kU^k)_{n-k})$. Elementary-divisor conjecture. The critical group

$$K(V(U^kD^k)_n)=K\left(\operatorname{Ind}_{A\wr\mathfrak{S}_{n-k}}^{A\wr\mathfrak{S}_n}\mathbbm{1}\right)$$

is given, as a list of elementary divisors, by

$$\left(1^{p_{n-k}},\left(r^{k}\frac{n!}{(n-k)!}\right)^{p_n-2p_{n-k}+p_{n-2k}},r^{k}\frac{n!}{(n-k)!}e_i\right),$$

where exponents involving rank sizes denote multiplicities. This predicts the precise elementary-divisor structure of these critical groups in the wreath-product differential tower.