Leopoldt conjecture for p-adic Lie extensions

Let $K_\infty/K$ be a $p$-adic Lie extension with Galois group $G$, let $V$ be a Galois representation, and let $T$ be a $\mathbb{Z}_p$-lattice in $V$. The equivalent conditions of Proposition

concern the corresponding global Iwasawa cohomology and hold independently of the choice of $T$. **Leopoldt conjecture for $p$-adic Lie extensions.** The equivalent conditions of Proposition

hold for some, and hence every, $\mathbb{Z}_p$-lattice $T$ in $V$. This is the weak Leopoldt conjecture used throughout the paper; the source notes that it is known in many cases for $K(\mu_{p^\infty})$, but leaves the assertion for arbitrary $V$ as conjectural.