The TKT a.1 defect-zero exclusion conjecture for quadratic fields

Let $K$ be a quadratic field, and let $G=\operatorname{Gal}(\mathrm{F}_3^2(K)\mid K)$ be its second $3$-class group. The TKT a.1 defect-zero exclusion conjecture. The transfer kernel type $\mathrm{a.1}$ cannot occur when the defect is $k=0$, that is, when $G$ is an infinitely capable vertex on the mainline of the coclass graph $\mathcal{G}(3,1)$. This predicts a restriction on the possible second $3$-class groups of quadratic fields; the supplied text gives no resolution status.