Beilinson's regulator conjecture for smooth projective curves

Let $X$ be a smooth, projective, geometrically connected curve of genus $g$ defined over $\mathbb{Q}$. Let $K_4^{(3)}(X)$ denote the third Adams eigenspace of $K_4(X)\otimes\mathbb{Q}$, let $r_B$ be the regulator map

$$r_B:K_4^{(3)}(X)\longrightarrow H^1(X(\mathbb{C}),\mathbb{R}(2))^+,$$

and let $H_1(X(\mathbb{C}),\mathbb{Q})^+$ be the fixed part under complex conjugation. Beilinson's conjecture. The following statements hold: $K_4^{(3)}(X)$ is a $\mathbb{Q}$-vector space of dimension $g$; and, if $(\Xi_1,\ldots,\Xi_g)$ is a $\mathbb{Q}$-basis of $K_4^{(3)}(X)$ and $(\gamma_1,\ldots,\gamma_g)$ is a $\mathbb{Q}$-basis of $H_1(X(\mathbb{C}),\mathbb{Q})^+$, then

$$L^{(g)}(X,-1)=c\pi^{-2g}\det\left(\int_{\gamma_i}r_B(\Xi_j)\right)_{1\leq i,j\leq g}$$

for some $c\in\mathbb{Q}^\times$. This is the $n=3$ case of Beilinson's conjecture, relating regulators in motivic or $K$-theory to special values of the curve's $L$-function; the source does not provide a resolution status.