The central zonotopal algebra conjecture for connected graphs

Let $G_1$ and $G_2$ be connected graphs. Their central zonotopal algebras are denoted by $4\mathcal{C}_{G_1}^{\mathcal{C}}4$ and $4\mathcal{C}_{G_2}^{\mathcal{C}}4$. For a graph, its bridge-free matroid is the graphical matroid obtained after deleting all bridges. The central graphical-algebra conjecture. The following are equivalent:

• $4\mathcal{C}_{G_1}^{\mathcal{C}}4$ and $4\mathcal{C}_{G_2}^{\mathcal{C}}4$ are isomorphic as non-graded algebras;
• $4\mathcal{C}_{G_1}^{\mathcal{C}}4$ and $4\mathcal{C}_{G_2}^{\mathcal{C}}4$ are isomorphic as graded algebras;
• the bridge-free matroids $M_{G_1}$ and $M_{G_2}$ are isomorphic.

This conjecture is the central analogue of the established classification for external graphical algebras; the source gives no resolution of the central case.