Deligne's conjecture for the classical exceptional series

Let $\mathsf{Exc}_{R,\lambda}$ be the classical exceptional diagram category. A rigid symmetric monoidal category is a symmetric monoidal category in which every object has a dual, and a Cauchy completion is an additive and idempotent completion. Deligne's conjecture. There exist a localization $R$ of $\mathbb Q[\lambda]$ at finitely many polynomials, a rigid symmetric monoidal category $\mathsf{Exc}^{Del}_{R,\lambda}$ with non-negative integers as objects and finitely generated $R$-modules as morphism spaces, and a symmetric monoidal functor $\operatorname{eval}:\mathsf{Exc}_{R,\lambda}\to\mathsf{Exc}^{Del}_{R,\lambda}$ that is the identity on objects and surjective on morphisms. After base extension along every map $\varphi:R\to S$, the resulting category has, for an open dense subset $U\subset\mathbb C$, semisimple Cauchy completion, dimensions $1,0,1,1,5,16,80$ in boundary degrees $0,1,2,3,4,5,6$, respectively, and at the listed special parameters its negligible quotient has Cauchy completion equivalent to representations of the corresponding supergroup. This is the proposed categorical realization of the exceptional series; the source presents it as a version of Deligne's conjecture incorporating the exceptional relation.