The equivalence conjecture for oriented dagger and disk-like higher categories

Let $SO_n$-dagger categories be the higher categories equipped with the oriented tangential structure described in the source, and let disk-like $n$-categories be the structures introduced by Morrison and Walker in blob-homology. The equivalence conjecture for oriented dagger and disk-like higher categories. There is an equivalence between $SO_n$-dagger categories and disk-like $n$-categories. This is motivated by the established equivalence between disk-like 2-categories and pivotal bicategories, together with the corresponding two-dimensional description of $SO_2$-dagger categories; the higher-dimensional equivalence remains open.