Classification conjecture for finite-dimensional graded *-simple algebras

Let $G$ be a finite abelian group and let $\widetilde{F}$ be an algebraically closed field of characteristic zero. For a finite-dimensional $G$-graded algebra $A$ with graded involution, write $A$ as $*$-graded simple when it has no nonzero proper graded ideals invariant under the involution. Let $H$ be a subgroup of $G$, let $\zeta:H\times H\to\mathbb{Q}[\sqrt[\mathfrak{m}]{1}]^*$ be a 2-cocycle, and set $\mathfrak{m}=|G|$. Define $\widetilde{F}^{\zeta}[H]$ as the corresponding twisted group algebra, $\widetilde{C}^{(1)}=M_k(\widetilde{F}^{\zeta}[H])$, and $\mathcal{B}=M_k(\widetilde{F}^{\zeta}[H])$.

Classification conjecture. The algebra $A$ is $*$-graded simple if and only if it is isomorphic as a graded $*$-algebra either to the $G$-graded simple algebra $\widetilde{C}^{(1)}$ with an elementary involution, or to $\widetilde{C}^{(2)}=\mathcal{B}\times\mathcal{B}^{op}$ with the exchange involution $\bar{*}$, where $\mathcal{B}^{op}$ is the opposite algebra and $\zeta$ has values in the algebraic extension of $\mathbb{Q}$ obtained by adjoining a primitive $\mathfrak{m}$-th root of unity.

This is the classification assumption referred to in the surrounding text. The paper states that it is known for cyclic groups of prime order and for the group of order $4$, while the general finite abelian case is presented as an assumption and remains open in the supplied text.