Waldhausen's chromatic reformulation of the Lichtenbaum–Quillen conjecture

Let $\mathcal{O}$ be a nice ring, let $\ell$ be prime, and let $\mathbb{S}/\ell$ denote the mod-$\ell$ sphere spectrum. Let $L_1^f$ denote finite-height localization, let $v_1$ be the chromatic periodicity class, and let $L_{K(1)}$ denote localization at Morava K-theory $K(1)$. Waldhausen's conjecture. The map

$$\operatorname{K}(\mathcal{O},\mathbb{Z}/\ell)\to L_1^f\operatorname{K}(\mathcal{O})\otimes\mathbb{S}/\ell\simeq \operatorname{K}(\mathcal{O},\mathbb{Z}/\ell)[v_1^{-1}]\simeq L_{K(1)}(\operatorname{K}(\mathcal{O}))\otimes\mathbb{S}/\ell$$

is an equivalence on large degrees. This is Waldhausen's topological reformulation of the Lichtenbaum–Quillen conjecture, connecting algebraic K-theory with chromatic localization; the supplied text does not state whether the conjecture is resolved for all nice rings.