The neighborhood expansion conjecture for positive entropic curvature on graph spaces

Let $G=({\mathcal X},E)$ be a graph endowed with the counting measure. For $z\in {\mathcal X}$, let $S_2(z)$ denote the vertices at graph distance two from $z$, and for $W\subset S_2(z)$ let $]z,W[$ denote the set of vertices adjacent to at least one element of $W$ along a shortest path from $z$. Neighborhood expansion conjecture. If for all $z\in {\mathcal X}$ and all $W\subset S_2(z)$,

$$\big|]z,W[\big|>\big|W\big|,$$

then the graph space $G$ has positive entropic curvature. This is presented as a natural sufficient condition for positive entropic curvature, motivated by the preceding necessary expansion property; the supplied text does not indicate whether the conjecture has been resolved.