Tangential-structure Floer module conjecture

Let $X$ be a Liouville domain with stable tangent bundle classified by $X \to BU$, and let $\Phi \to BU$ be a based space over $BU$. Suppose the classifying map $X \to BU$ factors through $\Phi \to BU$, and let $A:= \operatorname{Thom}(\Omega^2\Phi \to BU \times \mathbb{Z})$. Let $\phi$ be a symplectomorphism preserving the $\Phi$-structure. Tangential-structure Floer module conjecture. Floer theory of $\phi$, including Hamiltonian Floer theory and symplectic cohomology, can be defined as a module over the ring spectrum $A$; write $SH(\phi;A)$ or $SH(X;A)$ for this $A$-module. The preceding Floer moduli-space construction gives the relevant $\Omega^2\Phi$-structures, and the claim extends the framed and complex-oriented flow-category framework to symplectic cohomology with tangential structures.