Mayer's refinement of Scholz's conjecture for ramified dihedral cubic fields

Let $L$ be a non-Galois totally real cubic field with ramified Galois closure $N$ over a real quadratic field $K$, with conductor $f>1$. Let $U_N$ be the unit group of $N$, let $U_0$ be the subgroup generated by the units of the proper subfields, and suppose $N$ is of type $\alpha$, equivalently $U_N=U_0$. Let $T_{K,N}:\operatorname{Cl}_3(K)\to\operatorname{Cl}_3(N)$ be the transfer homomorphism of $3$-classes, defined by

$$T_{K,N}:\mathfrak{a}\cdot\mathcal{P}_K\longmapsto(\mathfrak{a}\mathcal{O}_N)\cdot\mathcal{P}_N.$$

Write $s$ for the number of prime divisors of $f$ that split in $K$, and let $\varrho_3(K)$ denote the $3$-class rank of $K$.

Mayer's conjecture. Non-Galois totally real cubic fields with these properties should exist in each of the following situations: type $\alpha_1$ with $\dim_{\mathbb{F}_3}(\ker(T_{K,N}))=2$, $\varrho_3(K)=2$, and $s=0$; type $\alpha_2$ with $\dim_{\mathbb{F}_3}(\ker(T_{K,N}))=1$, $\varrho_3(K)=1$, and $s=1$; or type $\alpha_3$ with $\dim_{\mathbb{F}_3}(\ker(T_{K,N}))=0$, $\varrho_3(K)=0$, and $s=2$.

This refines Scholz's conjecture by predicting existence in three ramified cases distinguished by the capitulation-kernel dimension, the $3$-class rank, and the number of split conductor primes. The source presents these as the three subtypes of type $\alpha$.