Polytope characterization of induced subgraphs with nonzero Euler characteristic

Let $G$ be a graph with no induced cycles of length divisible by three. An induced subgraph is a subgraph obtained by taking a vertex set of $G$ and all edges between those vertices. A polytope is the convex hull of finitely many points in a Euclidean space, and its reduced Euler characteristic is denoted by $\tilde{\chi}$.

Polytope characterization conjecture. There is a polytope $P$ whose vertices are labelled by the edges of $G$ and whose faces are in bijection with the induced subgraphs of $G$ having reduced Euler characteristic $\pm1$. That is, the vertices of a face of $P$ correspond to the edges of an induced subgraph of $G$.

The source presents this as a conjectural connection with edge polytopes. It gives the complete-bipartite graph example as motivation, but does not state a resolution of the general claim.