Boundedness conjecture for wild étale kernels in cyclotomic towers

Let $F$ be a totally real number field, and let $L/F$ be abelian and contain the $2\ell$-th roots of unity. Write

$$F_\infty=\bigcup_{n\in\mathbb N}F_n,$$

and

$$L_\infty=\bigcup_{n\in\mathbb N}L_n$$

for the cyclotomic $\mathbb Z_\ell$-extensions. Let $W\!K_{4i}(F_n)$ and $W\!K^-_{4i+2}(L_n)$ denote the indicated wild étale kernels and their imaginary components.

Combined Iwasawa–Greenberg conjecture. The following groups have orders bounded independently of $n$ and $i$: (i) the $\ell$-kernels $W\!K_{4i}(F_n)$; and (ii) the imaginary components $W\!K^-_{4i+2}(L_n)$.

This is stated as a consequence of combining the vanishing-$\mu$ conjecture of Iwasawa with Greenberg's conjecture. The source does not provide a general proof beyond the hypotheses under which it is derived.