The extremal root path conjecture for Schur σ-groups

Let $p$ be a prime and let $G$ be a finite non-abelian $p$-group. Its root path in the descendant tree of $G/G^{\prime}$ is obtained by iterated quotients by the last non-trivial terms of the lower central series. For a parent $W$, the nuclear rank $\nu(W)$ determines the possible step sizes of edges leaving $W$. In the descendant tree $\mathcal{T}(C_3\times C_3)$ of the abelian $3$-group $C_3\times C_3$, let

$$G=\operatorname{Gal}(\mathrm{F}_3^\infty(K)/K)$$

be the Galois group of the $3$-class field tower of an imaginary quadratic field $K$ with elementary $3$-class group $\operatorname{Cl}_3(K)\simeq C_3\times C_3$, and let $c=\operatorname{cl}(G)$.

Extremal root path conjecture. The root path $(\pi^{i-1}(G)\to\pi^i(G))_{1\leq i\leq c-2}$ consists of edges of maximal step size, equal to the nuclear rank: for every $1\leq i\leq c-2$,

$$\pi^{i-1}(G)\to\pi^i(G)\text{ is an edge of step size }s_i=\nu(\pi^i(G)).$$

Here $3\leq c<\infty$ is the nilpotency class of $G$. The conjecture asserts that the tower group's path through the descendant tree is extremal at every stage; the supplied text gives no resolution status.