Atiyah–Hitchin approximation conjecture for foliated topological gravity on exotic R4

Let $e$ be an exotic $\mathbb{R}^{4}$ belonging to the DeMichelis–Freedman radial family, and let the foliated topological limit of general relativity (FTL of GR) on $e$ be evaluated through its gravitational Euclidean path integral. Let ${\cal M}_{3}\times\mathbb{R}$ denote the fully anisotropic squashed 3-sphere end associated with the Atiyah–Hitchin (AH) gravitational instanton. Atiyah–Hitchin approximation conjecture. There exists such an exotic $\mathbb{R}^{4}$, $e$, for which the FTL of GR on $e$ is represented by the geometric end given by the AH instanton, in the sense that its contribution approaches the dominant contribution from $e$ to the gravitational Euclidean path integral; equivalently, $Z_{e}^{FTL}$ is approximated by the contribution from the AH instanton, despite the AH instanton not being an exotic geometry at the end of $e$. The conjecture proposes a local semiclassical description of the exotic end by the AH instanton, with the dominant path-integral contribution controlled by the quasi-modular, fully anisotropic geometry. The paper does not provide an explicit construction or proof, so the conjecture remains open.