The completion of the classification of involutive two-valued groups

In the 2022 work of V. M. Buchstaber, A. P. Veselov and A. A. Gaifullin an almost full classification of involutive commutative two-valued groups was obtained. In the 2024 work of A. A. Gaifullin the commutativity of an arbitrary involutive two-valued group was proved. The only question that remained open after these publications is the following: Is it true that if two infinitely generated abelian groups $A$ and $B$ are nonisomorphic, then two-valued coset groups $A/_{\pm \mathrm{Id}}$ and $B/_{\pm \mathrm{Id}}$ are also nonisomorphic as two-valued groups? We positively answer to this question.