The cyclotomic ideal-power conjecture for Fermat-type elements

Let $p$ be a prime number with $p>3$, let $K=\mathbb{Q}(\zeta)$ where $\zeta$ is a primitive $p$th root of unity, and put

$$\mathfrak p=(1-\zeta)\mathbb{Z}[\zeta].$$

For nonzero relatively prime integers $x,y$, consider the ideal equation

$$(x+y\zeta)\mathbb{Z}[\zeta]=\mathfrak p\mathfrak z^p\quad\text{or}\quad \mathfrak z^p,$$

according as $x+y\equiv0\pmod p$ or not, where $\mathfrak z$ is an ideal of $K$ prime to $\mathfrak p$.

The cyclotomic ideal-power conjecture. This equation has no solution except in the trivial cases

$$x+y\zeta=\pm(1-\zeta)\quad\text{or}\quad \pm(1+\zeta).$$

This conjecture is suggested by the preceding computations and by proofs of Fermat's Last Theorem in particular cases; its resolution is not indicated in the source.