Sangroniz's finiteness conjecture for finite 2-groups with odd real class number

Let $r$ be an odd natural number. A finite group element is real if it is conjugate to its inverse, and a conjugacy class is real if all its elements are real. Sangroniz's conjecture. For each odd natural number $r$, there are only finitely many finite $2$-groups with exactly $r$ real conjugacy classes. The conjecture seeks to explain why most small $2$-groups have an even number of conjugacy classes; finiteness is known for $r=1,3,5,7$ and, in the pro-$2$ setting, for odd $r\leq 23$, but the general finite-group statement remains open.