The stable homotopy type determines the move-equivalence class of a framed flow category

Let $(\mathscr{C}_1,\iota_1,\varphi_1)$ and $(\mathscr{C}_2,\iota_2,\varphi_2)$ be framed flow categories. Their associated stable homotopy types are the stable homotopy types determined by the corresponding realizations $\mathcal{X}(\mathscr{C}_1)$ and $\mathcal{X}(\mathscr{C}_2)$. Stable homotopy type conjecture. If the two framed flow categories determine the same stable homotopy type, then they are move equivalent to one another. The preceding corollary establishes this when the reduced homology with coefficients in $\mathbb{Z}/k\mathbb{Z}$ is supported in two consecutive degrees, while the general assertion is suggested as a stronger statement.