The K(1)-local Anderson duality conjecture for locally compact K-theory

Let $K$ denote non-connective $K$-theory. Let $p$ be an odd prime, let $F$ be a number field, and let $S$ be a finite set of finite places of $F$ such that $\frac{1}{p}\in\mathcal{O}_{S}$. Write $\mathsf{LC}\mathcal{O}_{S}$ and $\mathsf{LC}\mathcal{O}_{S,ad}$ for the locally compact categories appearing in the fiber sequences, and let $I_{\mathbf{Z}_{p}}$ denote the $p$-adic Anderson dual. The K(1)-local Anderson duality conjecture. There is an equivalence of fiber sequences

\xymatrix{ L_{K(1)}K(\mathcal{O}_{S}) \ar[r] \ar@{=>}[d] & L_{K(1)}K(\mathsf{LC}\mathcal{O}_{S,ad}) \ar[r] \ar@{<=>}[d] & L_{K(1)}K(\mathsf{LC}\mathcal{O}_{S}) \\ I_{\mathbf{Z}_{p}}L_{K(1)}K(\mathsf{LC}\mathcal{O}_{S}) \ar[r] & I_{\mathbf{Z}_{p}}L_{K(1)}K(\mathsf{LC}\mathcal{O}_{S,ad}) \ar[r] & I_{\mathbf{Z}_{p}}L_{K(1)}K(\mathcal{O}_{S}) \ar@{=>}[u] }

where the lower row is the Anderson dual of the top row, perhaps up to sign. Here an equivalence of fiber sequences means that the underlying bi-Cartesian square of spectra, including the null homotopy, is equipped with an equivalence to the dual square. The conjecture refines the established objectwise $K(1)$-local duality and remains unproved because making the equivalence of the full fiber sequences, including its coherences, is difficult.