The $b7$-Leopoldt conjecture for $b7$

Let $K$ be the CM field, let $b7$ be the chosen CM type, let $H$ be the finite cyclic extension of $K$ cut out by the character $c7$, and let $G=\operatorname{Gal}(H/K)$. Write $\mathfrak{o}_H^{\times}[\chi]=e_\chi(\mathbf C_p\mathfrak{o}_H^{\times})$ and $Y_{H,\Sigma}=\mathbf Z[\Sigma_H]$, where $\Sigma_H$ is the set of embeddings $H\hookrightarrow\mathbf C$ extending those in $b7$. The $p$-adic regulator map is

$$\log_{\Sigma,p}:\mathfrak{o}_H^{\times}[\chi]\longrightarrow Y_{H,\Sigma}[\chi].$$

The $b7$-Leopoldt conjecture. The map $\log_{\Sigma,p}$ is injective. This is a $b7$-component form of Leopoldt's conjecture and is used to control the non-vanishing of the relevant $p$-adic regulator; its status is open in the stated generality.