Σ-Leopoldt conjecture for the CM field L

Let $T$ be the relevant $p$-adic representation, let $e_\chi$ denote the $\chi$-idempotent, and let $H^1_f(G_{k_v},T)$ be the finite local cohomology subgroup at $v$. $\Sigma$-Leopoldt conjecture. The canonical map

$$e_\chi(\mathbb Z_p\otimes_{\mathbb Z}\mathcal O_L^\times)\longrightarrow\bigoplus_{v\in\Sigma}H^1_f(G_{k_v},T)$$

is injective. This is the Leopoldt-type injectivity condition needed for the subsequent $\mathcal L$-invariant discussion. The supplied context gives no resolution evidence.