Hamiltonian actions in Floer theory

Assume the setting of the Lie monoid action conjecture in Floer theory, with the actions Hamiltonian. Let $\mathcal{W}(T^*G)$ and $\mathcal{W}(T^*H)$ denote the wrapped Fukaya categories of the cotangent bundles, and let $\varphi_*$ and $\psi_*$ be the induced functors. Hamiltonian actions in Floer theory. The wrapped Fukaya categories $\mathcal{W}(T^*G)$ and $\mathcal{W}(T^*H)$ are $d$-curved $d$-categories; $\varphi$ and $\psi$ induce $d$-functors

$$\varphi_*\colon\mathcal{W}(T^*G)\to\mathcal{W}(T^*G'),\qquad \psi_*\colon\mathcal{W}(T^*H)\to\mathcal{W}(T^*H');$$

the whole co-category $\mathcal{F}uk(M)$, including non-invariant Lagrangian immersions, is a $d$-curved $(\mathcal{W}(T^*G),\mathcal{W}(T^*H))$ $u$-bimodule $d$-category; and $\Phi_\Lambda$ upgrades to a $(\varphi_*,\psi_*)$-equivariant functor of $u$-bimodule $d$-categories. The statement extends the preceding Floer conjecture to Hamiltonian actions and non-invariant immersed Lagrangians; the paper presents it as an expected wrapped-Fukaya-theoretic structure.