The commutative loop conjecture for integer spaces of sigma-sets

Let $A\in 2^{\mathds{N}}$, and let $3^{A}$ be the integer space equipped with the fusion operation $\oplus$. The conjecture concerns closure, an identity element, unique annihilating inverses, and commutativity.

Commutative loop conjecture. The structure $(3^{A},\oplus)$ satisfies:

1. $(\forall X,Y\in 3^{A})(X\oplus Y\in 3^{A})$;
2. $(\exists!\,\emptyset\in 3^{A})(\forall X\in 3^{A})(X\oplus\emptyset=\emptyset\oplus X=X)$;
3. $(\forall X\in 3^{A})(\exists!\,X^{-}\in 3^{A})(X\oplus X^{-}=X^{-}\oplus X=\emptyset)$;
4. $(\forall X,Y\in 3^{A})(X\oplus Y=Y\oplus X)$.

For $A=\{1,2\}$, the source verifies these four properties but also observes that $\oplus$ is not associative. The conjecture extends the verified algebraic properties to arbitrary $A\in2^{\mathds{N}}$, while associativity is not claimed.