Pivotal fusion categories admit combed field theory structures

Let $\mathcal C$ be a pivotal fusion category over a field of characteristic zero. An $SO(2)$ homotopy fixed point structure on $\mathcal C$ is the additional structure corresponding, under the cobordism hypothesis, to an $SO(2)$-structured local field theory; such a theory is called combed. Pivotal fusion category conjecture. Every pivotal fusion category in characteristic zero admits the structure of an $SO(2)$ homotopy fixed point, and therefore provides the structure of a combed 3-dimensional local field theory. This conjecture asserts that the pivotal structure upgrades the evident weaker homotopy fixed point structure to a full $SO(2)$ structure; the supplied text gives no resolution status.