be computed by counting holomorphic discs inside a symplectic manifold $M(\Sigma)$ associated to the surface $\Sigma$, with boundary conditions in two Lagrangian submanifolds $L(H_0), L(H_1) \subset M(\Sigma)$ associated to the handlebodies. The symplectic manifold suggested by this procedure is the moduli space of flat connections on the trivial $SU(2)$-principal bundle over $\Sigma$ introduced in [@AtiyahBott], and the Lagrangian submanifolds correspond to connections that extend flatly to $H_0$, resp. $H_1$.

Although the symplectic manifold $M(\Sigma)$ is not smooth (which is one of the main difficulties in defining Floer homology inside it), Huebschmann and Jeffrey proved that it can be realized as the symplectic quotient of a smooth finite dimensional $SU(2)$-Hamiltonian manifold, [@jeffrey], [@huebschmann], called the extended moduli space. Manolescu and Woodward then managed to define homology groups, replacing the symplectic manifold appearing in Atiyah’s suggestion by an open subset of this moduli space. These groups, called Symplectic Instanton homology (HSI) [@MW], are defined for every closed oriented three-manifold, and their isomorphism type is a topological invariant.

As a variant of the Atiyah-Floer conjecture, Manolescu and Woodward conjecture that these groups (with ${\mathbb{Q}}$-coefficients) are isomorphic to a variation of Instanton homology defined by Donaldson, [@MW Conjecture 1.2], and ask whether or not 