Flow categories as perfect bimodules conjecture

Let $\Theta$ be a tangential structure and let $R$ be the ring spectrum defined from it. Let $\operatorname{Flow}^{\Theta}$ denote the category of $\Theta$-oriented flow categories, and let $\operatorname{Perf}({}_R\operatorname{mod}_R)$ denote the category of perfect $R$-$R$ bimodules. Flow-category bimodule conjecture. There is an equivalence between $\operatorname{Flow}^{\Theta}$ and the homotopy category of perfect $R$-$R$ bimodules, $\operatorname{Perf}({}_R\operatorname{mod}_R)$. The one-object computation identifies morphism groups with the corresponding bordism groups, motivating a categorified Pontryagin–Thom model; the equivalence remains conjectural in the supplied text.