De-equivariantized generalized Haagerup modular data for cyclic even parts

Let $A$ be an abelian group with $A_e\cong\mathbb{Z}_{2^{n+1}}$. Cyclic-even-part de-equivariantization conjecture. The modular data of the Drinfeld center of the $\mathbb{Z}_2$-de-equivariantization of the generalized Haagerup category for $A$ are the stated modular data with $G=\mathbb{Z}_{2^n}^2\times A_o\times\widehat{A_o}$, $q_{1,e}(x,y)=\zeta_{2^{n+1}}^{x^2-y^2}$, and $q_{1,o}(a,\chi)=\chi(a)$. The source reports this for $A=\mathbb{Z}_4$ and $A=\mathbb{Z}_8$; the general claim remains open.