The set-partition conjecture for simple transitive 2-representations

Let $n\geq 1$, and let $\mathcal{C}_n$ be the left cell $2$-subcategory of projective functors for the star algebra considered in the paper. A simple transitive $2$-representation of $\mathcal{C}_n$ is a simple transitive action of this $2$-category on a finitary category. The set-partition conjecture. Equivalence classes of simple transitive $2$-representations of $\mathcal{C}_n$ are in bijection with set partitions of $\\{1,2,\ldots,n\\}$. The authors state that they are unable to classify all simple transitive $2$-representations in the general case; the conjecture proposes a complete classification indexed by set partitions.