Finiteness conjecture for 2-generated finite 2-groups with prescribed odd real class number

Let $k$ be an odd number. A finite $2$-group is 2-generated if it can be generated by two elements, and a real conjugacy class is a conjugacy class consisting of elements conjugate to their inverses. The 2-generator finiteness conjecture. For each odd number $k$, there are only finitely many finite $2$-groups generated by two elements with exactly $k$ real conjugacy classes. The authors motivate this problem by observing that the minimal pro-$2$ groups with an odd number of real conjugacy classes arise from Sylow subgroups of groups associated with division algebras and have at least three generators; the proposed two-generator finiteness remains open.