The feeble $\ell$-adic regulator conjecture

Let $K$ be an algebraic number field and let $K_\infty$ be its cyclotomic $\mathbb Z_\ell$-extension, with intermediate fields $K_n$ satisfying $[K_n:K]=\ell^n$. For an extension $K_m/K_n$, let $R_\ell(K_m/K_n)$ be the relative $\ell$-adic regulator formed from a system of fundamental relative units.

Feeble $\ell$-adic regulator conjecture. There exists an index $n_0$, depending only on $K$ and $\ell$, such that

$$R_\ell(K_m/K_n)\neq 0$$

for every $m>n\geq n_0$.

This is presented as a relative, eventual nonvanishing analogue of the $\ell$-adic regulator conjecture. The supplied text gives no resolution, so it remains open.