The Atiyah–Floer conjecture for pillowcase homology

Let $(X,K)$ be a knot in a homology $3$-sphere, and choose a $2$-tangle decomposition $(X,K)=(Y,T)\cup(D,U)$ as in the paper. Let $P$ denote the pillowcase, let $R_\pi(Y,T)$ be the perturbed traceless character variety, and let $L_1:R_\pi(Y,T)\to P$ be the restriction map. For suitably small generic perturbations $\pi$, an Atiyah–Floer conjecture asserts that $L_1$ is a restricted immersed $1$-manifold and that

$$H^\natural(Y,T,\pi)\cong I^\natural(X,K)$$

as relatively $\mathbb Z/4$-graded groups. This predicts that the pillowcase Lagrangian-Floer theory recovers reduced singular instanton homology; it holds in the examples computed in the paper but remains open in general.