Open-closed Landau-Ginzburg TFT axioms for compact critical locus

Let $(X,W)$ be a B-type open-closed topological Landau-Ginzburg theory, with $X$ a non-compact Calabi–Yau manifold and $W$ a holomorphic function whose critical set $Z_W$ is compact. Let $\bigl(\mathrm{HPV}(X,W),\mathrm{HDF}(X,W),\mathrm{Tr},\mathrm{tr},e\bigr)$ denote the cohomological bulk algebra, cohomology category, bulk trace, boundary traces, and bulk-boundary maps constructed from the differential models. The open-closed TFT conjecture. The system

$$\bigl(\mathrm{HPV}(X,W),\mathrm{HDF}(X,W),\mathrm{Tr},\mathrm{tr},e\bigr)$$

is a TFT datum and hence defines a quantum open-closed TFT in the axiomatic sense. Moreover, the boundary-bulk maps of this TFT datum are induced by the cochain-level boundary-bulk maps, and the corresponding bulk-boundary maps are induced in the same way. The paper proves all other defining properties on cohomology, while non-degeneracy of the bulk and boundary traces and the topological Cardy constraint remain to be established; the conjecture is motivated by path-integral arguments and known results for Stein $X$ with finitely many critical points.