The deep transfer conjecture for 3-class tower groups of totally real dihedral fields

Let $F=\mathbb{Q}(\sqrt{d})$ be a real quadratic field with fundamental discriminant $d>1$, $3$-class group $\operatorname{Cl}_3(F)\simeq C_3\times C_3$, shallow transfer kernel type $\mathrm{a}.1$, $\varkappa_s(F)=(0,0,0,0)$, and ground-state transfer target type $\tau(F)\sim\lbrack (9,9),(3,3)^3\rbrack$. Let $E_2,E_3,E_4$ be the three unramified cyclic cubic relative extensions of $F$ whose $3$-class groups have type $(3,3)$. For $2\leq i\leq 4$, define

$$\varkappa_d(F)_i=\#\ker(T_{F_3^{(1)}/E_i})$$

and let

$$\mathcal{G}_i=G_3^\infty{E_i}=\operatorname{Gal}((E_i)_3^{(\infty)}/E_i).$$

Each $E_i/\mathbb{Q}$ is a totally real dihedral extension of degree $6$. The deep transfer conjecture. For each $2\leq i\leq 4$, the connection between the component $\varkappa_d(F)_i$ of the deep transfer kernel type and the $3$-class tower group $\mathcal{G}_i$ is

$$\varkappa_d(F)_i=3\quad\Longleftrightarrow\quad\mathcal{G}_i\simeq\langle 243,27\rangle\text{ with }\varkappa_s(\mathcal{G}_i)=(1,0,0,0),$$

and

$$\varkappa_d(F)_i=9\quad\Longleftrightarrow\quad\mathcal{G}_i\simeq\langle 243,26\rangle\text{ with }\varkappa_s(\mathcal{G}_i)=(0,0,0,0).$$

The conjecture describes how deep transfer data in real quadratic fields determine the 3-class tower groups of associated totally real dihedral fields; the parser supplies no evidence resolving it, so its status remains open.