Zagier's conjecture on polylogarithms

Statement

Let $F$ be an algebraic number field with $r_1$ real embeddings, $r_2$ pairs of complex conjugate non-real embeddings and discriminant $d_F$, and let $m \ge 2$. Let $d_m = r_1 + r_2$ if $m$ is odd and $d_m = r_2$ if $m$ is even. Let $\sigma_1, \ldots, \sigma_{d_m}$ be the embeddings $F \to \C$ consisting of one from each complex pair, together with the real embeddings when $m$ is odd. Let $\operatorname{Li}_k$ be the polylogarithm of order $k$. Zagier's single-valued polylogarithm $L_m \colon \C \to \R$ is $$L_m(z) = \Re_m \Big( \sum_{k=0}^{m-1} \frac{2^k B_k}{k!} (\log \abs{z})^k \operatorname{Li}_{m-k}(z) \Big) \quad (\abs{z} \le 1).$$ It extends to a continuous function on $\C$ with $L_m(1/z) = (-1)^{m-1} L_m(z)$ for $z \ne 0$. Here $\Re_m$ is the real part for $m$ odd and the imaginary part for $m$ even. The $B_k$ are the Bernoulli numbers, with $B_1 = -1/2$. Extend $L_m$ additively to the free abelian group $\Z[F]$ on the elements of $F$. Then there are $\xi_1, \ldots, \xi_{d_m} \in \Z[F]$ and $q \in \Q^\times$ with $$\zeta_F(m) = q \, \frac{\pi^{e}}{\sqrt{\abs{d_F}}} \det\big(L_m(\sigma_j(\xi_i))\big)_{1 \le i, j \le d_m},$$ where $e = m r_2$ for $m$ odd and $e = m(r_1 + r_2)$ for $m$ even, and $\zeta_F$ is the Dedekind zeta function.