The lift of the weak proper Calabi–Yau structure on the Rabinowitz Fukaya category

Let $X$ be the setting of Theorem, let $\mathcal{RW}(X)$ denote its Rabinowitz Fukaya category, and let the weak proper Calabi–Yau structure on $\mathcal{RW}(X)$ be the one supplied by that theorem. A strong proper Calabi–Yau structure is a chain map

$$\tilde{\phi}: \operatorname{CC}_{*}(\mathcal{RW}(X),\mathcal{RW}(X))_{hS^{1}} \to {\mathbf{k}}[-n]$$

such that its composition with the canonical projection

$$\operatorname{CC}_{*}(\mathcal{RW}(X),\mathcal{RW}(X)) \to \operatorname{CC}_{*}(\mathcal{RW}(X),\mathcal{RW}(X))_{hS^{1}}$$

is the given weak proper Calabi–Yau structure. Lift conjecture. The weak proper Calabi–Yau structure from Theorem admits a lift to a strong proper Calabi–Yau structure on $\mathcal{RW}(X)$. This asks for an enhancement from the established weak structure to a homotopy-orbit-level, or strong, proper Calabi–Yau structure; the supplied text gives no resolution status.