The feeble conjecture on the 2-adic regulator for some 2-extensions

For an algebraic number field $K$ that is a finite 2-extension of the CM-field $k$ with trivial Iwasawa invariant $\mu_2(k)$, we prove that its cyclotomic $\mathbb Z_\ell$-extension $K_\infty/K$ satisfies the feeble conjecture on the 2-adic regulator [1]. In particular, this conjecture holds for $K_\infty/K$ if $K$ is a 2-extension of a field $k$ that is Abelian over $\mathbb Q$. We also obtain other results in the same direction.