Conjectural classification of even perfect and norm-perfect integers in higher cyclotomic fields

Let $\mathcal{R}$ be the set of cyclotomic fields over $\mathbb{Q}$ with class number $1$ generated by a primitive $p$-th root of unity, where $p$ is a rational prime or $4$. Let

$$K=\mathbb{Q}(\zeta_p)\in\mathcal{R},$$

with $p\neq 2,4$, and let $\zeta_p$ be chosen so that $1-\zeta_p$ is positive in $\mathbb{Z}[\zeta_p]$.

Classification conjecture for higher cyclotomic fields. The even perfect integers in $\mathbb{Z}[\zeta_p]$ are exactly

$$-\zeta_p^{-1}(1-\zeta_p)^{k-1}\bigl((1-\zeta_p)^k-1\bigr),$$

for $k\equiv 1\pmod{4p\mathbb{Z}}$ such that $(1-\zeta_p)^k-1$ is prime. The even norm-perfect integers are the associates of these perfect integers together with the associates of

$$(1-\zeta_p)^{k-1}\overline{(1-\zeta_p)^k-1},$$

for $k\equiv -1\pmod{4p\mathbb{Z}}$ such that $(1-\zeta_p)^k-1$ is prime.

The conjecture extends the described classifications in $\mathbb{Q}(i)$ and $\mathbb{Q}(\omega)$ to the remaining class-number-one cyclotomic fields under consideration. The supplied text gives no evidence that this proposed classification has been proved or disproved.