Iwasawa's cyclicity conjecture for eigenspaces of the class group

Let $l$ be an odd prime, let $A$ be the $l$-torsion part of the class group of $\mathbb Z[\mu_l]$, and let $\omega$ be the Teichmüller character of $G(\mathbb Q(\mu_l)/\mathbb Q)$. For $i\geq 0$, write $A^{[i]}=e_{\omega^i}A$ for the corresponding eigenspace. Iwasawa's conjecture. The group

$$A^{[l-1-n]}$$

is cyclic for every odd integer $n$ with $1\leq n<l-1$. This is the classical Iwasawa cyclicity conjecture; the supplied text does not state whether it has been resolved.