6 problems

A pro-$2$ Galois group is the Galois group of the maximal pro-$2$ extension of a field; a pro-$2$ group is ETG$_2$ if it can be constructed from $\mathbbm{Z}/2\mathbbm{Z}$, infinit…

Nuclear-rank conjecture. Every $2$-special group has nuclear rank $0$ or $2$, and its valid children are always $2$-special.

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

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

Let $F$ be a field of characteristic different from $2$. A finitely generated pro-$2$ group is weak elementary type (WET) when its mod-$2$ cohomology algebra can be obtained by a f…

Fourth-layer compositum conjecture.