Conjecture of capitulation for totally real number fields

Let $K$ be any totally real number field, let $p$ be a prime, and let ${\mathcal H}_K$ be its $p$-class group of exponent $p^e$. For an integer $N\geq e$, consider primes $\ell$ satisfying $\ell\equiv 1\pmod{2p^N}$ and the cyclotomic extension $K(\mu_\ell)$. Conjecture of capitulation. There exist infinitely many primes $\ell\equiv 1\pmod{2p^N}$, with $N\geq e$, such that ${\mathcal H}_K$ capitulates in $K(\mu_\ell)$. The conjecture predicts frequent capitulation in auxiliary cyclic $p$-extensions without any splitting assumption on $\ell$; numerical computations in the paper support the assertion, but no resolution is stated.