Generalized Yoshida conjecture for maximal CM subfields

Let $K$ be a finite abelian extension of a totally real field $F$ containing a CM-subfield, and let $K_{\mathrm{CM}}$ be its maximal CM-subfield. Put $G:=\operatorname{Gal}(K/F)$, and let $\mathfrak f_{K/F}\mid\mathfrak f_0$ and $\operatorname{Art}_{\mathfrak f_0}$ be as defined in the associated ray-class construction. For each $c$, let $X(c)$ be Yoshida's invariant, and let $p_{K_{\mathrm{CM}}}$ be Shimura's period symbol, with restriction $\tau|_{K_{\mathrm{CM}}}$ and the transfer map $\operatorname{Inf}^{-1}$. The generalized Yoshida conjecture. For every $\tau\in G$,

$$\prod_{c\in\operatorname{Art}_{\mathfrak f_0}^{-1}(\tau)}\exp(X(c))\bmod\overline{\mathbb Q}^{\times}=\pi^{\zeta_{\mathfrak f_0}(0,\tau)}p_{K_{\mathrm{CM}}}\left(\tau|_{K_{\mathrm{CM}}},\operatorname{Inf}^{-1}\left(\sum_{\sigma\in G}\zeta_{\mathfrak f_0}(0,\sigma)\sigma\right)\right).$$

This extends Yoshida's CM-period conjecture from CM-fields to abelian extensions having a CM-subfield. It is motivated by the equivalent formulation of the original conjecture and remains conjectural in the general setting.