Factorizable Lie bialgebra conjectures for equivariant Rabinowitz homology

Let $Q$ be a manifold and let $W$ be a Liouville domain. Write $\widehat{\mathbb{H}}_*^{S^1}\Lambda$ for Rabinowitz string homology, $S\mathbb{H}_*^{S^1}({\partial}W)$ for $S^1$-equivariant Rabinowitz Floer homology, and use bars for the corresponding reduced homology groups. Factorizable Lie bialgebra conjectures. (i) The graded Lie bialgebras $\widehat{\mathbb{H}}_*^{S^1}\Lambda$ and, more generally, $S\mathbb{H}_*^{S^1}({\partial}W)$ are factorizable. (ii) If $Q$ is orientable and $\dim Q\geq 4$, the graded factorizable Lie bialgebra on $\widehat{\mathbb{H}}_*^{S^1}\Lambda$ is isomorphic to the one on the double of $\overline{\mathbb{H}}_*^{S^1}\Lambda$. (iii) If $W$ is a strongly essential Weinstein domain of dimension $2n\geq 8$, the graded factorizable Lie bialgebra on $S\mathbb{H}_*^{S^1}({\partial}W)$ is isomorphic to the one on the double of $\overline{S\mathbb{H}}_*^{S^1}(W)$. These claims propose an equivariant analogue of the non-equivariant double construction and would provide factorizable graded Lie bialgebra structures in string topology and symplectic homology. The supplied text gives no resolution evidence for these statements, so they remain open.