The realization conjecture for finite 2-special groups as narrow tower groups

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

Narrow realization conjecture. Every finite $2$-special $2$-group arises as the narrow tower group of some cyclic cubic field.

This conjecture identifies the class of groups on which a distribution for narrow $2$-class tower groups should be placed; the paper investigates it computationally using IPADs, but does not establish it in general.