Dimension reduction conjecture for representations of motion groups of links with generalized axes

A link $L\subset\mathbb{S}^3$ has a generalized axis $\gamma$ with fiber surface $F$ when the complement of $\gamma$ fibers over $\mathbb{S}^1$ with fiber $F$. A $1$-extended $(3+1)$-TQFT and a $1$-extended $(2+1)$-TQFT have labels from which their associated representations are constructed. Dimension reduction conjecture. There is a map from labels of a $1$-extended $(3+1)$-TQFTs to those of $1$-extended $(2+1)$-TQFTs such that the representation of the motion group of $L$ from the $(3+1)$-TQFT decomposes as a direct sum of representations of some surface braid groups of $F$. This conjecture proposes an explicit reduction of motion-group representations arising from $(3+1)$-dimensional Dijkgraaf–Witten TQFTs to surface braid-group representations arising from $(2+1)$-dimensional theories; the supplied text does not indicate a resolution.