The principal capitulation conjecture for unramified abelian extensions

Let $K/k$ be an unramified abelian extension of number fields in which all infinite places of $k$ split completely. Write $Cap_{K/k}$ for the capitulation kernel, consisting of ideal classes of $k$ that become principal in $K$.

Principal capitulation conjecture. The order of the capitulation kernel is a multiple of the degree of the extension:

$$[K:k]\mid |Cap_{K/k}|.$$

This extends the preceding theorem for cyclic extensions, where the same divisibility is proved under the stated splitting condition at infinity. The conjecture is suggested by decomposing the abelian Galois group into cyclic factors; its general validity for abelian, not necessarily cyclic, extensions is left open here.