The tower ground state conjecture for selected imaginary quadratic fields

Let $K=\mathbb{Q}(\sqrt d)$ be an imaginary quadratic field with fundamental discriminant $d$, and let $\ell_3(K)$ denote the length of its $3$-class field tower. Let $S=\mathrm{G}_3^\infty(K)$ denote the tower group, and use $\operatorname{lo}(S)$, $\operatorname{cl}(S)$ and $\operatorname{cc}(S)$ for its logarithmic order, nilpotency class and coclass.

Tower ground state conjecture. The fields with discriminants $d\in\{-225\,299,-343\,380,-423\,476,-486\,264\}$ of type $\mathrm{F}.7$, $d\in\{-27\,156,-241\,160,-477\,192,-484\,804\}$ of type $\mathrm{F}.11$, $d=-291\,220$ of type $\mathrm{F}.12$, and $d\in\{-167\,064,-296\,407,-317\,747,-401\,603\}$ of type $\mathrm{F}.13$ have $3$-class field towers of exact length

$$\ell_3(K)=3$$

with a suitable Schur $\sigma$-group as tower group. For all four types, the tower group satisfies

$$\operatorname{lo}(S)=20,\qquad \operatorname{cl}(S)=9,\qquad \operatorname{cc}(S)=11,$$

with $\zeta_1(S)=(9,9)$ or $(9,3,3)$, $\gamma_2^2(S)=(27,27,9,3,3,3)$ or $(27,9,9,9,3,3)$, and usually $\#\operatorname{Aut}(S)=2\cdot3^{25}$, rarely $2\cdot3^{26}$. The supplied text gives no resolution status.