Gross's regulator conjecture for imaginary quadratic extensions

Let $k$ be an imaginary quadratic field, let $F/k$ be a finite abelian extension with Galois group $G$, let $r<0$, and let $onzerochar\chi\in\widehat{G}$ be a character. Write $K_{1-2r}(O_F)_{/\mathrm{tors}}$ for the odd-degree higher algebraic $K$-group modulo torsion, let $\rho_F^r$ be the Beilinson regulator, let $L_S(s,\chi)$ be the $S$-truncated Artin $L$-function, and set

$$w_{1-r}(F^{\mathrm{ker}(\chi)}):=\left|H^0\left(\mathrm{Gal}(\overline{\mathbb{Q}}/F^{\mathrm{ker}(\chi)}),\mathbb{Q}/\mathbb{Z}(1-r)\right)\right|.$$

Also let $e_\chi=|G|^{-1}\sum_{\sigma\in G}\chi^{-1}(\sigma)\sigma$ be the idempotent associated with $\chi$. Gross's conjecture. There exists a unique element $\epsilon(\chi,S)\in K_{1-2r}(O_F)_{/\mathrm{tors}}\otimes_{\mathbb{Z}}\mathbb{Z}[\chi]$ such that

$$\rho_F^r(\epsilon(\chi,S))=w_{1-r}(F^{\mathrm{ker}(\chi)})L_S'(r,\chi^{-1})|G|e_\chi$$

in $\mathbb{R}[G]\otimes_{\mathbb{Z}}\mathbb{Z}[\chi]$. This is a regulator formulation of Gross's conjecture, relating special Artin $L$-values to higher $K$-theory. Its status is not resolved in the supplied text.