Localization Dichotomy Conjecture for non-periodic systems

Let $H$ be a Hamiltonian satisfying the assumptions of Proposition, and let $P$ be its spectral projector onto $\sigma_0$. A generalized Wannier basis is a basis of $\operatorname{range}(P)$ consisting of generalized Wannier functions; it is exponentially localized if its elements have exponential localization, and $s$-localized if they have localization exponent $s$. For a projection $P$ on $L^2(\mathbb{R}^2)$, let $C(P)$ denote its Chern marker, when the limit exists:

$$C(P):= \lim_{L \rightarrow \infty} \frac{2 \pi i}{4 L^2} \operatorname{tr}{\left( \chi_{L} P \Big[ [ X, P ], [ Y, P ] \Big] P \chi_{L}\right)}.$$

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

This conjecture extends the localization-topology correspondence from periodic to non-periodic systems, asserting that exponential localization, $s$-localization at the threshold $s=1$, and vanishing of the Chern marker are equivalent. The supplied text does not state whether the conjecture has been resolved.