The filtration conjecture for algebraic K-theory of endomorphisms

Let $A$ be the ring under consideration, and let $\widetilde{K}(\operatorname{End}(A))$ denote the reduced algebraic $K$-theory spectrum of its category of endomorphisms. Let $V^\bullet$ denote the Verschiebung filtration, and let $\operatorname{THH}(A)$ and $\operatorname{TR}(A)$ denote topological Hochschild homology and topological restriction homology, respectively. The filtration conjecture. There is a filtration $V^\bullet$ on $\widetilde{K}(\operatorname{End}(A))$ such that the first associated graded component is $\operatorname{THH}(A)$. The completion with respect to the filtration is $\operatorname{TR}(A)$. This proposes a higher algebraic $K$-theoretic explanation of the relationship between Witt vectors, topological Hochschild homology, and topological restriction homology; the paper states that these conjectures are not proved and are intended for future work.