Abouzaid–Blumberg's truncated flow-category equivalence conjecture

Fix an integer $n$. Let $\Flow_{\tau_{\leq n}}$ be the category of flow categories with morphisms given by homotopy classes of $\tau_{\leq n}$-morphisms, where $\tau_{\leq n}$ denotes truncation of a spectrum with respect to the standard $t$-structure. Abouzaid–Blumberg conjecture. The category $\Flow_{\tau_{\leq n}}$ is equivalent to the category whose objects are perfect $MO$-modules and whose morphisms $M\to N$ are homotopy classes of $\tau_{\leq n}MO$-linear maps

$$M\otimes_{MO}\tau_{\leq n}MO\longrightarrow N\otimes_{MO}\tau_{\leq n}MO.$$

Moreover, this equivalence should be compatible, in an appropriate sense, with the equivalence of the untruncated perfect-$MO$-module conjecture. This conjecture proposes a truncated stable-homotopy model for flow-category morphisms.