The rank 1 abelian Stark conjecture for differenced ray class zeta functions

Let $K$ be a real quadratic field and let $\{\rho_1,\rho_2\}$ be its real embeddings. Let $\cc$ be an ideal of the maximal order $\OO_K$, let $S=\{\rho_2\}$, and let $A$ be a ray ideal class modulo $\cc S$. Define the ray class zeta function

$$\zeta(s,A)=\sum_{\aa\in A}N(\aa)^{-s}$$

and, for the element

$$R=\{a\OO_K:a\con -1\Mod{\cc}\text{ and }a\text{ is positive at }\rho_2\},$$

define the differenced ray class zeta function $Z_A(s)=\zeta(s,A)-\zeta(s,RA)$. If $R$ is not the identity of $\Cl_{\cc\infty_2}(\OO_K)$, then the Stark conjecture. There is an algebraic unit $\e_A$ generating the ray class field $L_{\cc\infty_2}$ corresponding to $\Cl_{\cc\infty_2}(\OO_K)$ such that

$$Z_A'(0)=\log(\rho_1(\e_A)).$$

Moreover, the units are compatible with the Artin map:

$$\e_{\id}^{\Art(A)}=\e_A.$$

This is the rank 1 abelian case of Stark's conjectures, relating derivatives at zero of ray class zeta functions to logarithms of algebraic units in ray class fields. The stated compatibility describes the Galois action through the Artin map.