Gras's p-rationality conjecture for number fields

Let $K$ be a number field, let $S_p$ denote the set of places above $p$, and let ${\mathcal T}_{S_p}$ be the group used in the source. A number field $K$ is $p$-rational when $G_{S_p}$ is free.

Gras's conjecture. For $p\gg 0$, one has

$${\mathcal T}_{S_p}=1,$$

and consequently $K$ is $p$-rational for $p\gg 0$.

The source says this follows from the Leopoldt conjecture and presents it as a fundamental conjecture due to Gras; no resolution status is supplied.