The finitude of tamely ramified pro-$p$ extensions of number fields with cyclic $p$-class groups

Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In this work, we study the finitude of the Galois group $G_{\{\mathfrak{q}\}}(F)$ of $F_{\{\mathfrak{q}\}}$ over $F$. We prove that $G_{\{\mathfrak{q}\}}(F)$ is finite for the majority of $\mathfrak{q}$'s such that the generator rank of $G_{\{\mathfrak{q}\}}(F)$ is two, provided that for $p=3$, $F$ is not a complex quartic field containing the primitive third roots of unity.