The cardinality conjecture for integer spaces of sigma-sets

Let $A$) be a sigma-set with $|A|=n$. The associated integer space is denoted by $3^{A}$, and $2^{A}$ and $2^{A^{-}}$ are combined using the operation $\oplus$ to form $\left\langle 2^{A},2^{A^{-}}\right\rangle$.

Cardinality conjecture.

$$\left|3^{A}\right|=\left|\left\langle 2^{A},2^{A^{-}}\right\rangle\right|=3^{n}.$$

The preceding computations establish the formula for sigma-sets of cardinalities through five, while the conjecture asserts it for arbitrary finite cardinality $n$.