Flow-category Thom spectrum equivalence

Let $F$ be a space and let $R:= \operatorname{Thom}(\Omega F \to BO \times \mathbb{Z})$ be the associated unital ring spectrum. Let $\operatorname{Flow}^F$ denote the stable $\infty$-category of flow categories built from manifolds with $\Omega F$-structure. Flow-category Thom spectrum conjecture. There is an equivalence between $\operatorname{Flow}^F$ and the $\infty$-category of left $R$-modules. When $R$ is associative but not commutative, analogous versions of $\operatorname{Flow}^F$ should correspond to right $R$-modules and $R$-bimodules. In the framed case, $R=\mathbb{S}$ and the assertion was proved in the cited work; the general left/right/bimodule models are constructed conjecturally in the paper.