The weak Leopoldt conjecture for the representation $T$

Let $K$ be a number field, let $K_\infty/K$ be a $\mathbb{Z}_p$-extension, let $\mathcal{K}'/K$ be a finite extension, and let $\mathcal{K}=\mathcal{K}'K_\infty$. For each $E\in\Omega(\mathcal{K}'/K)$, write $\Lambda=\mathcal{R}[[\operatorname{Gal}(K_\infty/K)]]$ and let

$$H^2_{\mathrm{Iw}}(\mathcal{O}_{E,S(E)},T^*(1))= \mathop{\mathpalette\varlim@{\leftarrowfill@\scriptscriptstyle}}\nmlimits@ _n H^2(\mathcal{O}_{E_n,S(E_n)},T^*(1)).$$

Weak Leopoldt conjecture. For every $E\in\Omega(\mathcal{K}'/K)$, $H^2_{\mathrm{Iw}}(\mathcal{O}_{E,S(E)},T^*(1))$ is a torsion $\Lambda$-module.

This is the weak Leopoldt condition used in the Iwasawa-theoretic study of Euler systems; it controls the second Iwasawa cohomology group over the cyclotomic tower. The supplied text assumes the conjecture but gives no resolution status.