Cyclotomic Kummer splitting criterion implying SFLT2

Let $p>3$ be prime and let $(p,u,v)\in\mathbb Z^3$ with $u$ and $v$ coprime and $p\mid v$. For a $p$-principal prime $q$ with $q\nmid puv$, let $n$ be the order of $v/u$ modulo $q$, set $\xi=e^{2\pi i/n}$, and let $M=\mathbb Q(\xi,\zeta)$. Define $\mathfrak q=(q,u\xi-v)$. Cyclotomic Kummer splitting criterion. If every such triple admits a $p$-principal prime $q$ for which $\mathfrak q$ is not totally split in

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

then SFLT2 holds. The conjecture is intended to turn a failure of SFLT2 into a simultaneous total-splitting condition in Kummer extensions; the source gives no resolution status.