Conjecture on asymptotically CAT(0) graphs and hyperbolicity

A graph is called $\delta$-CAT(0) when its geodesic triangles satisfy the corresponding CAT(0) comparison condition up to an additive constant, and it is asymptotically CAT(0) when its asymptotic geometry is CAT(0). A graph is $\delta$-hyperbolic when its geodesic triangles are uniformly thin. Asymptotically CAT(0) graph conjecture. A graph is asymptotically CAT(0) if and only if it is $\delta$-hyperbolic. The source motivates this as an analogue of the fact that a graph is $\delta$-CAT(0) if and only if it is hyperbolic, but gives no resolution of the asymptotic version.