Gras's conjecture on the vanishing of tame ramification for large primes

Let $K$ be a number field, let $p$ be a prime, and let ${\mathcal T}_{S_p}$ denote the relevant tame ramification module for the maximal extension of $K$ unramified outside the places above $p$. Gras's conjecture. For every number field $K$, one has

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

for all sufficiently large primes $p$. This conjecture predicts that tame ramification eventually vanishes as the ramified prime varies; under Leopoldt's conjecture, it is equivalent in this setting to the freeness of the pro-$p$ group $G_{S_p}$ for sufficiently large $p$.