Near-group realization conjecture for the inverse-involution family

Let $A$ be an odd finite abelian group and let $G=\mathbb{Z}_{2^n}\times A$ with $n\geq1$. Inverse-involution near-group conjecture. The constructed modular data are the modular data of the Drinfeld center of the $\mathbb{Z}_2$-de-equivariantization of a near-group category for $\mathbb{Z}_{2^{n+1}}\times A$ with multiplicity $2^{n+1}|A|$. The statement is true for $n=1$ and trivial $A$, while the general infinite-family realization is open.