SFLT2 criterion via a nonsplit p-principal prime

Let $p>3$ be prime. For a triple $(p,u,v)\in\mathbb Z^3$ with $u$ and $v$ coprime and $p\mid v$, let $q$ be a $p$-principal prime with $puv\not\equiv0\pmod q$, let $n$ be the order of $v/u\pmod q$, let $\xi=e^{2\pi i/n}$, and let $\mathfrak q$ be a prime ideal above $q$ satisfying $u\xi-v\equiv0\pmod{\mathfrak q}$. Put $M=\mathbb Q(\xi,\zeta)$. SFLT2 criterion conjecture. If such a prime $q$ and ideal $\mathfrak q$ can always be found so that $\mathfrak q$ is not totally split in

$$M\left(\sqrt[p]{\langle\varepsilon_k\varepsilon_1^{-1}\rangle_{k=1,\ldots,p-2}}\right)/M,$$

then SFLT2 holds. This is proposed as a sufficient condition for the Strong Fermat's Last Theorem conjecture and is not assigned a resolution in the source.