The commutative gyrogroup conjecture

A gyrogroup is a set equipped with a binary operation and the associated gyroautomorphisms satisfying the gyrogroup axioms; it is commutative when its binary operation satisfies $a\oplus b=b\oplus a$ for all elements $a,b$. The commutative gyrogroup conjecture. If a gyrogroup is commutative, then it is a group. This would characterize commutative gyrogroups as ordinary groups; the source reports that computations suggest the claim, but provides no general proof or resolution.