Quilted Atiyah–Floer conjecture

Let the three proposed Floer field theories be extended, for appropriate $G$-bundles, to $2$-functors

$$\operatorname{Bor}_{2+1+1}\to\operatorname{Symp}^{G}_{2+1},\qquad \operatorname{Bor}_{2+1+1}\to\operatorname{SIn}_{2+1},\qquad \operatorname{Bor}_{2+1+1}\to\operatorname{DBor}_{2+1}.$$

Quilted Atiyah–Floer conjecture. The three extended Floer field theories induce isomorphic $2$-functors

$$\operatorname{Bor}_{2+1+1}\to\operatorname{Cat}.$$

This is the Donaldson-theoretic, categorified extension of the Atiyah–Floer conjectures from Heegaard splittings and cyclic Cerf decompositions. The source presents it as a proposal contingent on suitable bundle choices and, where necessary, resolving reducible connections.