Beilinson's conjecture for higher K-groups of smooth projective varieties

Let $X_{{\mathbb Q}}$ be a smooth projective variety over ${{\mathbb Q}}$, and let $X_{{\mathbb R}}:=X_{{\mathbb Q}}\times_{{\mathbb Q}}{{\mathbb R}}$. Let $K_i(X_{{\mathbb Q}})^{(j)}_{{\mathbb Z}}\subset K_i(X_{{\mathbb Q}})^{(j)}$ be the integral part, and let $j>0$ be an integer. Write $F_\infty$ for the real Frobenius action and $L(M,s)$ for the $L$-function of a motive $M$; write $L^*(M,m)$ for the leading Taylor coefficient at $s=m$, and let $\sim_{{{\mathbb Q}}^\times}$ denote equality up to multiplication by an element of ${{\mathbb Q}}^\times$. Beilinson's conjecture. The regulator map

$$\operatorname{reg}_{{\mathbb R}}:K_j(X_{{\mathbb Q}})^{(j)}_{{\mathbb Z}}\otimes{{\mathbb R}}\longrightarrow H^{j-1}_B(X_{{\mathbb R}},{{\mathbb R}}(j-1))^{F_\infty=1}$$

is bijective, and

$$\det[K_j(X_{{\mathbb Q}})^{(j)}_{{\mathbb Z}}]\otimes\det[H^{j-1}_B(X_{{\mathbb R}},{{\mathbb Q}}(j-1))^{F_\infty=1}]^{-1}\sim_{{{\mathbb Q}}^\times}L^*(h^{j-1}(X_{{\mathbb Q}}),0).$$

The order-of-vanishing assertion in the source is conditional on the functional equation. This is a higher-dimensional generalization of the classical Dirichlet regulator conjecture, and remains open in general.