The converse characterization of R-Poincaré duality spaces by graph cobordisms

Let $R$ be the coefficient ring and let $M$ be a compact space. Consider the analogous data $(\mathfrak{I}_{\hat\Psi},\mathfrak{t}_{\hat\Psi})$ from Definition, regarded as data in $((\operatorname{SPr}^L_R)^{\mathrm{l-lax}}_{\operatorname{Mod}_R/})^{2\simeq}$. An $R$-Poincaré duality space is a space satisfying the $R$-Poincaré duality condition used in the construction.

Converse characterization conjecture. If $(\mathfrak{I}_{\hat\Psi},\mathfrak{t}_{\hat\Psi})$ is a commutative Frobenius algebra in

$$((\operatorname{SPr}^L_R)^{\mathrm{l-lax}}_{\operatorname{Mod}_R/})^{2\simeq},$$

then $M$ is an $R$-Poincaré duality space.

The forward construction only uses that $M$ is an $R$-Poincaré duality space; an $R$-orientation is used only to simplify formulas. The conjecture asserts that the resulting commutative Frobenius algebra structure also detects Poincaré duality.