Independence of signature data for equivariant instanton complexes

Let $Y$ be a rational homology sphere, let $E$ be the bundle used to define the equivariant instanton complex, let $R$ be the coefficient ring, and let $\sigma$ be a choice of signature data. Write $\widetilde{CI}_*(Y,E,\sigma;R)$ for the resulting complex, equipped with its graded $C_*(SO(3);R)$-equivariant structure. Signature-data independence conjecture. The complex $\widetilde{CI}_*(Y,E,\sigma;R)$ is independent of the choice of signature data $\sigma$ up to graded $C_*(SO(3);R)$-equivariant quasi-isomorphism. The conjecture asserts that the signature data introduced to formulate admissibility do not affect the equivariant instanton complex up to the stated notion of equivalence for rational homology spheres.