Strong Fermat's Last Theorem conjecture

Let $p>3$ be prime, let $K=\mathbb Q(\zeta)$ be the $p$-th cyclotomic field, and let $u,v\in\mathbb Z\setminus\{0\}$ be coprime. The equation

$$(u+v\zeta)\mathbb Z[\zeta]=\mathfrak w_1^p\ \text{or}\ \mathfrak p\mathfrak w_1^p$$

according as $u+v\not\equiv0\pmod p$ or not, equivalently

$$N_{K/\mathbb Q}(u+v\zeta)=w_1^p\ \text{or}\ pw_1^p,\qquad w_1\in1+p\mathbb Z,$$

where $\mathfrak w_1$ is an ideal of $K$ prime to $p$, has no solutions except the trivial ones $u+\zeta v=\pm(1-\zeta)$ and $\pm(1+\zeta)$. Strong Fermat's Last Theorem conjecture. This is presented as a conjecture implying Fermat's Last Theorem; its resolution status is not specified in the source.