The nuclear-rank conjecture for 2-special groups

Let $G$ be a $2$-special $2$-group, and let its nuclear rank be the invariant used in the paper to describe its valid children. A valid child is a child in the valid-group tree described there.

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

Nuclear rank $0$ means that the group is childless. The conjecture is motivated by extensive computational searches among $2$-special groups, but no general proof is given.