The commutative loop conjecture for integer spaces

Let $IN=\{1,2,3,\ldots\}$ and let $3^X$ be the integer space associated with $X\in 2^{IN}$. Define an operation $\oplus$ by

$$A\oplus B\leftrightarrow A\cup B.$$

The commutative loop conjecture. For all $X\in 2^{IN}$, $(3^X,\oplus)$ satisfies: (1) $(\forall A,B\in 3^X)(A\oplus B\in 3^X)$; (2) $(\exists !\xi\in 3^X)(\forall A\in 3^X)(\xi\oplus A=A\oplus\xi=A)$; (3) $(\forall A\in 3^X)(\exists !A^{\star}\in 3^X)(A\oplus A^{\star}=A^{\star}\oplus A=\xi)$; and (4) $(\forall A,B\in 3^X)(A\oplus B=B\oplus A)$. These conditions assert closure, an identity element, unique inverses, and commutativity. The conjecture relates the algebraic structure of integer spaces to non-associative finite invertible loops (NAFILs). The paper gives examples satisfying the conditions but reports no general proof or disproof.