Finiteness conjecture for p-ramification invariants

Let $K$ be a number field satisfying the Leopoldt conjecture for every prime $p$, and let ${\mathcal T}_p$ be the torsion subgroup associated with the maximal Abelian $p$-ramified extension modulo the compositum of the $\mathbb{Z}_p$-extensions. p-ramification finiteness conjecture. The product of these invariants over all primes is finite:

$$\prod_p {\mathcal T}_p\text{ is finite}.$$

The paper relates this to $p$-rationality, Greenberg-type consequences, and cohomological finiteness, but gives no resolution status.