Localization dichotomy for non-periodic gapped systems

Let $P$ be an orthogonal projector that admits an exponentially localized kernel. A generalized Wannier basis for $P$ is an orthonormal basis of $\operatorname{range}(P)$; it is $s$-localized if there are center points $\boldsymbol{\mu}_{\alpha}\in\mathbb{R}^2$ and a finite constant $C$ such that

$$\int_{\mathbb{R}^2} \langle \boldsymbol{x}-\boldsymbol{\mu}_{\alpha}\rangle^{2s}|\psi_{\alpha}(\boldsymbol{x})|^2\,d\boldsymbol{x}\leq C$$

for every basis element, where $\langle \boldsymbol{x}-\boldsymbol{\mu}_{\alpha}\rangle=(|\boldsymbol{x}-\boldsymbol{\mu}_{\alpha}|^2+1)^{1/2}$. The Chern marker $C(P)$ is the thermodynamic-limit invariant defined for $P$.

Localization dichotomy conjecture. The following statements are equivalent: (a) $P$ admits a generalized Wannier basis that is exponentially localized; (b) $P$ admits a generalized Wannier basis that is $s$-localized for $s=1$; (c) $P$ is topologically trivial in the sense that its Chern marker $C(P)$ exists and equals zero.

This conjecture extends the periodic localization dichotomy to non-periodic gapped systems. It identifies exponential localization, finite second moment, and vanishing Chern marker as equivalent conditions; the supplied text gives no resolution status.