Atiyah–Floer conjecture for instanton and Lagrangian Floer homology

Let $M=H_g^1\cup_{\Sigma_g}H_g^2$ be a Heegaard decomposition of a homology $3$-sphere, and let $I(M)$ denote its instanton Floer homology. For the trivial $SU(2)$ bundles on the handlebodies, write $R(H_g^i;\mathcal E_{H_g})$ for their representation spaces, viewed as Lagrangian submanifolds of $R(\Sigma_g;\mathcal E_{\Sigma_g})$. Atiyah–Floer conjecture. The instanton Floer homology is isomorphic to the Lagrangian Floer homology

$$I(M)\cong HF(R(H_g^1;\mathcal E_{H_g}),R(H_g^2;\mathcal E_{H_g})).$$

This conjecture proposes a correspondence between gauge-theoretic instanton Floer homology and symplectic Lagrangian Floer homology associated with a Heegaard splitting. Its status is not specified in the source.