The realization conjecture for finite 2-select groups as wide tower groups

Let $K$ be a cyclic cubic field, and let a finite $2$-group be called 2-select as defined in the paper. Its wide $2$-class tower group is the corresponding wide tower group.

Wide realization conjecture. Every finite $2$-select $2$-group arises as the wide tower group of some cyclic cubic field.

The conjecture supplies the proposed support for a distribution of wide $2$-class tower groups. The paper gives computational evidence, while also observing restrictions on which viable narrow/wide group pairs occur; the general realization claim remains open.