The twisted equivariant cyclic-to-de Rham quasi-isomorphism

Let $G$ be a compact Lie group acting smoothly on a compact manifold $M$, let $\alpha\in H^3_G(M,\mathbb Z)$, let $H$ be the relevant transformation groupoid, and let $L$ be the associated equivariant flat line-bundle family. Write $\operatorname{PC}_\bullet^G(C_c^\infty(H,L))$ for the equivariant periodic cyclic complex with differentials $b$ and $\mathcal B$, and let $\mathcal A_G^{\bullet}(M,L)$ be the complex of global twisted equivariant differential forms with differential $d_{\mathrm{eq}}^\alpha$. The twisted de Rham conjecture. The family of chain maps $\{\tau_g\}_{g\in G}$ from Theorem main induces a quasi-isomorphism

$$(\operatorname{PC}_\bullet^G(C_c^\infty(H,L)),b+\mathcal B)\longrightarrow(\mathcal A_G^{\bullet}(M,L),d_{\mathrm{eq}}^\alpha).$$

This would provide a de Rham model for equivariant twisted K-theory, extending the untwisted equivariant cyclic-homology description. The source gives no resolution evidence, so the conjecture remains open.