Goncharov's conjecture on Sah algebra cohomology and algebraic K-theory

Let $\widetilde{\mathcal{P}}(S^{n-1})$ be the reduced spherical scissors congruence group, and let the direct sum

$$\bigoplus_{n\geq0}\widetilde{\mathcal{P}}(S^{n-1})$$

be equipped with the Sah algebra Hopf structure, whose coproduct is given by the Dehn invariants. Write $H^i\big(\bigoplus_{n\geq0}\widetilde{\mathcal{P}}(S^{n-1})\big)_n$ for its degree-$n$ Hopf algebra cohomology, and let $K_{2n-i}(\mathbb{C})^{(n)}$ denote the weight-$n$ part of the Adams decomposition of algebraic K-theory. Goncharov's conjecture. The Hopf algebra cohomology $H^i\big(\bigoplus_{n\geq0}\widetilde{\mathcal{P}}(S^{n-1})\big)_n$ is isomorphic to the positive eigenspace of the complex conjugation action on

$$K_{2n-i}(\mathbb{C})^{(n)}.$$

This conjecture connects the cohomology of the Hopf algebra arising from spherical scissors congruence and Dehn invariants with the Adams-weighted algebraic K-theory of $\mathbb{C}$.