The elliptic Stark conjecture for CM curves

Let $E$ be a CM elliptic curve, let $\chi$ be a character in the setting above, and assume the Dimension conjecture so that the Stark regulator ratio $A(E,\chi)$ is defined. Here $\mathbb Q(\chi)$ denotes the field generated by the values of $\chi$, and $A(E,\chi)^\sigma$ denotes the image under $\sigma\in\operatorname{Gal}(\mathbb Q(\chi)/\mathbb Q)$. The elliptic Stark conjecture. The quantity $A(E,\chi)$ belongs to $\mathbb Q(\chi)$, and for every $\sigma\in\operatorname{Gal}(\mathbb Q(\chi)/\mathbb Q)$,

$$A(E,\chi)^\sigma=A(E,\chi^\sigma).$$

This is the algebraicity and Galois-equivariance assertion for the Stark regulator ratio attached to the twisted $L$-function of a CM elliptic curve. The parser supplies no evidence resolving the conjecture, so its status is left open.