Canonical local identifications of transgressed flat bundles

Let $G$ be a compact Lie group acting smoothly on a compact manifold $M$, and let $b1\in H^3_G(M,\mathbb Z)$. For each $g\in G$, let $P^g\to M^g$ be the associated $G$-equivariant flat $S^1$-bundle. If $h=g\exp X\in G^g$ with $X\in\mathfrak g^g$ is sufficiently near $g$, then the canonical-identification conjecture. There exists a canonical isomorphism of flat $S^1$-bundles over $M^h$,

$$\phi_{gh}:P^h\xrightarrow{\cong}P^g|_{M^h},$$

where $P^g|_{M^h}\to M^h$ is the restriction of $P^g\to M^g$ along $M^h\subseteq M^g$. Such identifications are needed to assemble the local coefficient systems into a sheaf of global twisted equivariant differential forms and thereby define delocalized twisted equivariant cohomology. The source presents this as a proposed construction, and gives no evidence that the asserted canonical isomorphism has been established.