Gross's higher Stark conjecture

Let $L/K$ be a finite Galois extension of number fields with group $G$, let $S$ contain all archimedean places, and let $r>1$ be an integer. For a complex character $\chi$ of $G$, let $A_{\phi_{1-r}}^S(\chi)$ be the quotient of the regulator determinant $R_{\phi_{1-r}}(\chi)$ by the leading coefficient $L_S^{\ast}(1-r,\chi)$ of the $S$-truncated Artin $L$-series. Gross's conjecture. For every $\sigma\in\operatorname{Aut}(\mathbb{C})$ and every character $\chi$ of $G$,

$$A_{\phi_{1-r}}^S(\chi^{\sigma})=A_{\phi_{1-r}}^S(\chi)^{\sigma}.$$

This is Gross's higher analogue of Stark's conjecture, expressing an equivariance property of regulator quotients and Artin $L$-values under automorphisms of $\mathbb{C}$; its resolution is not specified in the source.