The Witt-group decomposition conjecture for trivial syllepsis

Let $A$ be a finite abelian group. Write $\mathcal{W}itt(A,\mathsf{triv})$ for the Witt group associated to $A$ with trivial syllepsis, let $\mathcal{W}itt$ denote the ordinary Witt group, and let $A[2]$ denote the relevant 2-stage construction on $A$. Witt-group decomposition conjecture. There is an isomorphism of groups

$$\mathcal{W}itt(A,\mathsf{triv})\cong \mathcal{W}itt \oplus H^4(A[2];\mathbb{C}^{\times}).$$

This conjecture is motivated by the observation that Witt spaces commute with equivariantizations. The supplied text gives no resolution status.