Amenability, exactness, and crossed products of the stable Higson corona

Let $Y$ be a proper metric space equipped with an isometric action of a countable discrete group $H$. Denote by $\mathfrak{c}^{\mathrm{red}}Y$ the reduced stable Higson corona of $Y$. Say that $H$ acts amenably on the Higson compactification of $Y$, and let $\rtimes_{\max}$ and $\rtimes_{\mathrm{red}}$ denote the maximal and reduced crossed products.

Amenability–exactness conjecture. The following conditions are equivalent:

1. $H$ acts amenably on the Higson compactification of $Y$.
2. $\mathfrak{c}^{\mathrm{red}}Y$ is an amenable $H$-$C^*$-algebra.
3. $H$ is exact and

$$\mathfrak{c}^{\mathrm{red}}Y\rtimes_{\max}H\cong \mathfrak{c}^{\mathrm{red}}Y\rtimes_{\mathrm{red}}H.$$

The conjecture seeks to characterize amenability of the reduced stable Higson corona action simultaneously through $C^*$-algebra amenability and through exactness together with equality of maximal and reduced crossed products. The supplied text describes it as a general conjecture and gives only partial results, so its resolution is not stated.