The converse conjecture for Postnikov–Shapiro algebras and bridge-free matroids

Let $G_1$ and $G_2$ be connected graphs, and let ${\mathcal B}_{G_1}^T$ and ${\mathcal B}_{G_2}^T$ denote their associated algebras. For a graph, its bridge-free matroid is the graphical matroid of the graph obtained by removing all bridges.

Converse conjecture. The algebras ${\mathcal B}_{G_1}^T$ and ${\mathcal B}_{G_2}^T$ are isomorphic if and only if the bridge-free matroids of $G_1$ and $G_2$ are isomorphic.

The forward implication is conjectured to provide a converse to the preceding result, which establishes that isomorphic bridge-free matroids imply isomorphic algebras. The source gives no resolution of the converse implication.