Binary-operation conjecture for the ternary power set of a sigma-set

Let $A$ be a

-set such that there \exists $A^{\ast}$, the

-antiset of $A$. Define

$$\oplus:3^{A}\times 3^{A}\rightarrow 3^{A},\qquad \oplus(x,y)=x\cup y.$$

Ternary power-set operation conjecture. The operation $\oplus$ is a binary operation on $3^{A}$.

This conjecture concerns whether union is closed on the ternary power set when a sigma-antiset of $A$ exists. The source provides no resolution, so its status remains open.