Tableaux problem 46

46.[Gaetz2018a]*Conj. 1.6 Let $\mathfrak{G}:\{e\}=G_{0}\subset G_{1}\subset G_{2}\subset\cdots$ be a tower of groups and let $r\in\mathbb{Z}_{>0}$. Define $P$ to be the splitting diagram of the representations of the tower of groups, with weights that are the multiplicities of the splittings. The tower is $r$-dual if $P$ is a self-dual graded graph with differential degree $r$. The tower is $r$-differential if the tower is $r$-dual and the weights of $P$ are all 1, it is a differential poset. Show that: (a) If $\mathfrak{G}$ is $r$-differential then $P(\mathfrak{G})\cong\mathbb{Y}^{\otimes r}$ and there exists an abelian group $A$ of order $r$, not depending on $n$, such that $G_{n}\cong A\wr S_{n}$ for all $n\geq 0$. (b) If $\mathfrak{G}$ is $r$-dual then $P(\mathfrak{G})\cong(d_{1}\mathbb{Y})\times\cdots\times(d_{k}\mathbb{Y})$ for some $d_{1},\ldots,d_{k}$ and there exists a group $H$ of order $r$, not depending on $n$, such that $G_{n}\cong H\wr S_{n}$ for all $n\geq 0$.