Extended Uniqueness Conjecture for coherent weakly approximable triangulated categories

Let $\ct$ be a coherent, weakly approximable triangulated category. Let $\cc_{\mathbf{Sp}}$ be the collection

$$\left\{\mathrm{Ho}(\mathbf{Sp}),\mathrm{Ho}(\mathbf{Sp})^b,\mathrm{Ho}(\mathbf{Sp})^-,\mathrm{Ho}(\mathbf{Sp})^+,\mathrm{Ho}(\mathbf{Sp})\SB,\mathrm{Ho}(\mathbf{Sp})^-_c,\mathrm{Ho}(\mathbf{Sp})^b_c,\mathrm{Ho}(\mathbf{Sp})^c\right\}$$

and let $\cc_\ct$ be the corresponding collection

$$\left\{\ct,\ct^b,\ct^-,\ct^+,\tsb,\ct^-_c,\ct^b_c,\ct^c\right\}.$$

Suppose $\ca_1\subset\cb_1$ in $\cc_{\mathbf{Sp}}$ and $\ca_2\subset\cb_2$ in $\cc_\ct$ are matching subcategories.

Extended Uniqueness Conjecture. If there is an exact equivalence

$$\ca_1\cong\ca_2,$$

then there is an exact equivalence

$$\cb_1\cong\cb_2.$$

This generalizes the Margolis Uniqueness Conjecture to matching pairs of intrinsic subcategories. Its validity is not established in general; the paper presents it as a broader open conjecture motivated by invariance results and the special cases available when enhancements exist.