BV double conjectures for Rabinowitz loop homology and symplectic homology

Let $Q$ be a manifold, let $W$ be a Liouville domain of dimension $2n$, and let $\widehat{\mathbb{H}}_*\Lambda$, $\overline{\mathbb{H}}_*\Lambda$, $S\mathbb{H}_*({\partial}W)$, and $\overline{S\mathbb{H}}_*(W)$ denote the Rabinowitz and reduced loop or symplectic homology objects appearing in the corresponding double constructions. Let $\lambda$ and $\eta$ be the coproduct and unit maps in an odd BV unital infinitesimal algebra. BV double conjectures. (i) If $H_1Q=0$ and $\dim Q\geq 4$, the BV operator on $\widehat{\mathbb{H}}_*\Lambda$ is isomorphic to the BV operator on the double of $\overline{\mathbb{H}}_*\Lambda$, so that the odd BV Frobenius algebra structures are isomorphic. (ii) If $H^{n-1}W=0$ and $2n\geq 8$, the BV operator on $S\mathbb{H}_*({\partial}W)$ is isomorphic to the BV operator on the double of $\overline{S\mathbb{H}}_*(W)$, so that the odd BV Frobenius algebra structures are isomorphic. (iii) The double construction for odd BV unital infinitesimal algebras generalizes to all such algebras; equivalently, the assumption $\lambda\eta=0$ in the cited proposition is redundant. (iv) With respect to this generalized double construction, the assumptions $H_1Q=0$ and $H^{n-1}W=0$ can be dropped from the two preceding double theorems, and their isomorphisms can be upgraded to isomorphisms of odd BV Frobenius algebras. These conjectures seek a BV enhancement of the double descriptions already known for the underlying odd Frobenius structures. The paper states that parts (i), (ii), and (iv) would follow from a BV enhancement of the cone formalism, while part (iii) should be informed by explicit formulas that do not require the vanishing assumptions; no resolution of the conjecture is supplied here.