The bibar-cube conjecture for perverse sheaves on symmetric powers

Let $A$ be a primitive bialgebra in a braided monoidal abelian category, and let ${\mathbb{B}\mathbb{B}}_n(A)$ denote its $n$th bibar-cube, viewed as a double representation of $\2^{n-1}$. Let $S$ be the diagonal factorization on $\operatorname{Sym}^n(\mathbb{C})$, and let $\operatorname{Perv}(\operatorname{Sym}^n(\mathbb{C}),S,\mathcal{V})$ be the category of perverse sheaves with coefficients in $\mathcal{V}$ smooth with respect to $S$. The bibar-cube conjecture. (a) The category $\operatorname{Perv}(\operatorname{Sym}^n(\mathbb{C}),S,\mathcal{V})$ is equivalent to a full subcategory of $\operatorname{Rep}^{(2)}(\2^{n-1})$ consisting of double representations satisfying an explicit set of relations (Rel). (b) The relations (Rel) are precisely the universal relations holding in all bibar-cubes ${\mathbb{B}\mathbb{B}}_n(A)$ for primitive bialgebras in braided monoidal abelian categories. This would identify the combinatorial double representations arising from primitive bialgebras with the perverse-sheaf description on the symmetric power, and would characterize the relations by their universal validity across all such bibar-cubes. The supplied text does not establish the result; it says that a precise result is given in the next chapter, so the resolution status of this candidate should be checked.