The delocalized twisted equivariant Chern character 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 $R(G)$ be the representation ring, and let $R^\infty(G)=C^\infty(G)^G$ be the algebra of smooth conjugation-invariant functions. Let $K^\bullet_{G,\alpha}(M)$ denote twisted equivariant K-theory and $H_{G,\mathrm{delocalized},\alpha}^\bullet(M)$ the delocalized twisted equivariant cohomology. The delocalized twisted Chern character conjecture. The equivariant Chern character, composed with the isomorphism induced by the twisted cyclic-to-de Rham quasi-isomorphism, gives a natural isomorphism

$$K^\bullet_{G,\alpha}(M)\otimes_{R(G)}R^\infty(G)\xrightarrow{\sim}H_{G,\mathrm{delocalized},\alpha}^\bullet(M).$$

This would identify delocalized twisted equivariant cohomology as a de Rham realization of twisted equivariant K-theory after extension of scalars from $R(G)$ to $R^\infty(G)$. The claim is presented as an immediate consequence of the preceding conjecture and is unresolved in the supplied text.