Fiber 2-functor conjecture for parameterized 2+1d non-chiral SPT phases

Let $\mathcal{C}$ be a fusion 2-category and let $2\operatorname{Vec}$ be the 2-category of finite semisimple 1-categories. A fiber 2-functor is a tensor 2-functor $F:\mathcal{C}\rightarrow2\operatorname{Vec}$. The fiber 2-functor conjecture. For non-chiral 2+1d SPT states labeled by $F$, $S^1$-parameterized families are classified by the group of isomorphism classes of monoidal natural automorphisms of $F$, while $S^2$-parameterized families are classified by the group of invertible monoidal modifications of the identity natural transformation of $F$. The conjecture extends the categorical description of SPT phases to parameterized families in higher dimensions; the paper gives consistency checks and examples, but the general claim remains open.