The FI-homology decomposition conjecture for injective cogenerators

Let $a,b\in\mathbb{N}$, let $n$ be a homological degree, and let $\mathfrak{Crit}(a,b)$ denote the set of critical pairs $(\lambda,\mu)$ used in the proposition. For each such pair, let $S^\lambda$ and $S^\mu$ be the corresponding irreducible symmetric-group representations, and let $|\mu/(\lambda\cap\mu)|$ denote the size of the indicated skew diagram. FI-homology decomposition conjecture. There is an isomorphism of $\mathfrak{S}_b\times\mathfrak{S}_a$-modules

$$H_n^\mathbf{FI}\bigl(\mathbb{k}\operatorname{Hom}_\mathbf{FI}(-,\mathbf{b})\bigr)(\mathbf{a})\cong\bigoplus_{\substack{(\lambda,\mu)\in\mathfrak{Crit}(a,b)\\|\mu/(\lambda\cap\mu)|=n}}S^\lambda\boxtimes S^\mu.$$

The proposition preceding this conjecture proves the displayed direct sum embeds in the FI-homology and gives equality when $n=0$; the conjecture asserts that no additional summands occur in higher homological degrees.