The cardinality conjecture for meta-spaces of sigma-sets

Let $A\in 2^{\mathds{N}^{0}}$ and $B\in 2^{\mathds{N}}$. The meta-space generated from the sigma-set fusion $A\oplus B$ and the corresponding fusion with the antiset $B^{-}$ is

$$\left\langle 2^{A\oplus B},2^{A\oplus B^{-}}\right\rangle.$$

Meta-space cardinality conjecture. For all such $A$ and $B$,

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

The example preceding the conjecture has $|A|=|B|=2$ and yields $36=2^{2}\cdot3^{2}$. The general formula is presented as a conjecture, with no resolution supplied in the source.