Local and global universal norms in the cyclotomic $\mathbb Z_\ell$-extension of an algebraic number field

For an algebraic number field $K$ and a prime $\ell$ we study the subgroups of global universal norms $U_{S,1}(K)$ and of everywhere locally universal norms $U_{S,2}(K)$ in the cyclotomic $\mathbb Z_\ell$-extension $K_\infty$ of $K$ in the pro-$\ell$-completion of the group of $S$-units $U_S(K)[\ell]$, where $S$ is the set of all places over $\ell$. Assuming that the $\ell$-adic Schanuel conjecture holds, we prove the finiteness of the index $(U_{S,2}(K):U_{S,1}(K))$, whence we obtain a conditional proof of a conjecture in [1] on the Iwasawa module.
We also obtain an unconditional proof of all these results in the particular case when $K$ is a Galois extension of $\mathbb Q$ with symmetric Galois group $G=S_4$, $K$ contains an imaginary quadratic field, and $\ell$ is a prime such that the decomposition subgroup of its prime divisor coincides with the Sylow $3$-subgroup of $G$.