The power-of-two quaternion-brace counting conjecture

Let a quaternion brace be a brace whose multiplicative group is isomorphic to a generalized quaternion group. Power-of-two quaternion-brace conjecture. There are seven isomorphism classes of quaternion braces of size $2^k$ for $k>4$.

This is presented as a consequence of the preceding quaternion-brace counting conjecture: for $m=2^{k-2}$ and $k>4$, one has $m\equiv0\bmod8$. Thus the asserted count is seven, while the supplied text gives no proof of the general preceding conjecture and only reports finite verification.