Capitulation and stability conjecture for p-class groups in cyclotomic extensions

Let $K$ be any real number field and let ${\mathcal H}_K$ be its $p$-class group, of exponent $p^{e(K)}$. For any prime number $\ell \equiv 1 \pmod {2p^N}$, with $N \geq e(K)$, let $K_n$ be the subextension of $K(\mu_\ell)$ of degree $p^n$ over $K$, where $n \in [0,N]$. Capitulation and stability conjecture. The following assertions hold: there exist infinitely many $\ell$ such that ${\mathcal H}_K$ capitulates in $K(\mu_\ell)$; there exist infinitely many $\ell$ such that the capitulation of ${\mathcal H}_K$ in $K(\mu_\ell)$ is due, for some $n \in [1,N]$, to

$$e(K_n) \in [1,n-s(K_n)]$$

when

$$m(K_n) \in [p^{s(K_n)},p^{s(K_n)+1}-1]$$

for some $s(K_n) \in [0,n-1]$; and, for $N \gg 0$, the stability case

$$\# {\mathcal H}_{K_{n_0+1}}=\# {\mathcal H}_{K_{n_0}}$$

for some $n_0<N$ occurs for infinitely many $\ell$. This is intended to extend the proposed capitulation principle and to provide a route toward the real abelian Main Conjecture; the source presents these assertions as conjectural and does not give a proof or resolution.