Flow-category Hochschild homology conjecture

Let $\Theta$ be a tangential structure, let $R$ be its associated ring spectrum, and suppose $\operatorname{Flow}^{\Theta} \simeq \operatorname{Perf}({}_R\operatorname{mod}_R)$. For $\mathcal{F} \in \operatorname{Flow}^{\Theta}$ corresponding to an $R$-bimodule $M$, let $R_{\Delta}$ be the diagonal bimodule, and let $\operatorname{Hom}_{R \otimes R^{op}}$ denote homotopy classes of bimodule maps. Flow-category Hochschild homology conjecture. There are functorial isomorphisms

$$\Omega_*^{\Theta}(\mathcal{F}) \simeq \pi_*(M \otimes_{R\otimes R^{op}} R_{\Delta}), \qquad \Omega^*_{\Theta}(\mathcal{F}) \simeq \operatorname{Hom}_{R\otimes R^{op}}(M,R_{\Delta}),$$

and, under these isomorphisms, capsule concatenation corresponds to the displayed composition landing in

$$\pi_*(R_{\Delta} \otimes_{R\otimes R^{op}} R_{\Delta}) =: \pi_*THH(R).$$

The one-object computations verify the expected formulas in basic cases, while the functorial identification for general flow categories and its compatibility with concatenation remain conjectural.