Chinburg's strong Stark conjecture

Let $K$ be a number field, let $L/K$ be a finite Galois extension with group $G$, let $S$ be a finite set of places of $K$ containing $S_{\infty}$ and all places ramified in $L/K$, and suppose that the finite places in $S$ generate the class group of $L$. Let $\chi$ be a character of $G$, let $\phi_S:X_{L,S}\to\mathcal{O}_{L,S}^{\times}$ be an injective $\mathbb{Z}[G]$-homomorphism inducing a rational isomorphism, and let $A_S(\chi,\phi_S)$ and $q_{\phi_S}(\check\chi)$ be the associated analytic ratio and $q$-index. The character field is $\mathbb{Q}(\chi)$. Strong Stark conjecture. For every $\sigma\in\operatorname{Aut}(\mathbb{C})$, one has

$$A_S(\chi,\phi_S)^\sigma=A_S(\chi^\sigma,\phi_S),$$

and

$$q_{\phi_S}(\check\chi)=(A_S(\chi,\phi_S))$$

in $\mathbb{Q}(\chi)$. This is Chinburg's refinement of Stark's conjecture: it identifies the principal ideal generated by the regulator-to-$L$-value ratio with an algebraically defined $q$-index. The source presents it as a conjecture for sufficiently large $S$; its general status is not specified here.