The Cayley characterization conjecture for GKZ-rational configurations

Let $A_0,A_1,\dots,A_r$ be vector configurations in $\mathbb{Z}^r$, and let $A$ be their Cayley configuration

$$A=\\{e_0\\}\times A_0\cup\cdots\cup\\{e_r\\}\times A_r\subset\mathbb{Z}^{r+1}\times\mathbb{Z}^r,$$

where $e_0,\dots,e_r$ is the standard basis of $\mathbb{Z}^{r+1}$. The configuration $A$ is essential if the Minkowski sum $\sum_{i\in I}A_i$ has affine dimension at least $|I|$ for every proper subset $I$ of $\\{0,\dots,r\\}$. A configuration is GKZ-rational when its associated GKZ hypergeometric system has the rationality property considered in the source. The Cayley characterization conjecture. An arbitrary configuration $A$ is GKZ-rational if and only if $A$ is affinely isomorphic to an essential Cayley configuration. This conjecturally gives a classification of GKZ-rational configurations; the paper proves necessary restrictions and establishes results for several classes, but supplies no general resolution.