Zagier's K-theoretic polylogarithm conjecture

Let $F$ be a number field and let $m\geq 2$. Write $K_{2m-1}(F)$ for the algebraic K-group, let $P_m$ be the modified polylogarithm, and let $R_m(F):K_{2m-1}(F)\longrightarrow{\mathbb R}^{n_{\mp}}$ be the Borel regulator, where $n_{\pm}$ is chosen according to the sign of $(-1)^m$ and $n_{\mp}$ is the other signature number. Let $\mathbb Z[F^{*}]$ denote the free abelian group on $F^{*}$, and let $B_m(F)$ be a finitely generated subquotient of this group.

Zagier's K-theoretic conjecture. There is a homomorphism $B_m(F)\longrightarrow K_{2m-1}(F)$ with finite kernel and cokernel such that the composition with $R_m(F)$ equals the map induced by

$$P_m(F):B_m(F)\longrightarrow{\mathbb R}^{n_{\mp}}.$$

Moreover, the image of $P_m(F)$ is a lattice whose covolume is

$$\frac{\zeta_F(m)\sqrt{d_F}}{\pi^{m n_{\pm}}}.$$

Beilinson constructed a map of the indicated type for a subquotient, and the conjecture is closely related to the Lichtenbaum conjecture. The latter predicts that the ratio of the orders of the kernel and cokernel is determined by torsion in $K_{2m-1}(\mathcal O_F)$, by $K_{2m-2}(\mathcal O_F)$, and by powers of $2$; the full assertion remains open.