Lemmermeyer's conjecture on 3-class groups of pure cubic fields

Let $p$ be a prime number such that $p\equiv 1\, (\mathrm{mod}\,3)$, let $L=\mathbb{Q}(\sqrt[3]{p})$ be the corresponding pure cubic field, and let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ be its normal closure. Write $C_{L,3}$ and $C_{\mathrm{k},3}$ for the $3$-components of the class groups of $L$ and $\mathrm{k}$, respectively, and let $\left(\frac{\cdot}{p}\right)_3$ denote the cubic residue symbol. Lemmermeyer's conjecture. The following hold: (1) $C_{L,3}$ is a cyclic $3$-group, and if it contains a cyclic subgroup of order $9$, then $p\equiv 1\,(\mathrm{mod}\,9)$. (2) If $p\equiv 4,7\,(\mathrm{mod}\,9)$, then

$$C_{\mathrm{k},3}\simeq\begin{cases} \mathbb{Z}/3\mathbb{Z} & \text{if } \left(\frac{3}{p}\right)_3\neq 1,\\ (\mathbb{Z}/3\mathbb{Z})^2 & \text{if } \left(\frac{3}{p}\right)_3=1. \end{cases}$$

(3) If $p\equiv 1\,(\mathrm{mod}\,9)$, then $\operatorname{rank} C_{\mathrm{k},3}\in\{1,2\}$, independently of the value of $\left(\frac{3}{p}\right)_3$. The conjecture gives a precise description of the relevant $3$-class groups and their ranks for pure cubic fields and their normal closures. The case $p\equiv 1\,(\mathrm{mod}\,9)$ was partially proved by Gerth in 2005, while the case $p\equiv 4,7\,(\mathrm{mod}\,9)$ remains open.