Leopoldt's abelian conjecture for totally real fields

Let $K$ be a totally real field and let $p$ be a prime. Let $\Delta$ be the Galois group of the maximal abelian pro-$p$ extension of $K$ unramified outside $p$. Leopoldt's abelian conjecture.

$$\dim_{\mathbb Q_p}\Delta[1/p]=1.$$

This is the abelian Leopoldt conjecture, equivalently asserting that the global units have maximal possible rank in the product of the $p$-adic local unit groups, or that the $p$-adic regulator is nonzero. The source gives no resolution status.