Character-wise minus Gross–Kuz'min conjecture

Let $K$ be a CM-field, let $p\ne2$, and let $R\subseteq K$ be a totally real subfield such that $K/R$ is normal. Set $G=\operatorname{Gal}(K/R)$ and assume $K\cap R_\infty=R$. Let $V_{A'}=A'\otimes_{\mathds{Z}_p}\overline{\mathds{Q}}_p$, let $\chi\in\operatorname{Ir}(G)$, and let $V_{A'}^{(\chi)}=\varepsilon_\chi\cdot V_{A'}$. If $T=\gamma-1$ for a topological generator $\gamma$ of $\Gamma$, let $f_{A',\chi}(T)$ be the characteristic polynomial of $T$ on $V_{A'}^{(\chi)}$. Character-wise minus Gross–Kuz'min conjecture. The characteristic polynomial $f_{A',\chi}(T)$ is not divisible by $T$. This character-wise formulation is equivalent to the minus Gross–Kuz'min conjecture for all relevant minus characters, and is open in general.