The central zonotopal algebra conjecture for matrices

Let $A_1\in\mathbb{R}^{n_1\times m_1}$ and $A_2\in\mathbb{R}^{n_2\times m_2}$ have ranks $n_1$ and $n_2$, respectively. A column is a bridge-column if deleting it decreases the rank. Let $k_i$ be the number of bridge-columns of $A_i$, and let $A_i'\in\mathbb{R}^{(n_i-k_i)\times(n_i-k_i)}$ be the submatrix obtained by deleting all $k_i$ bridge-columns and those $k_i$ rows such that $\operatorname{rk}(A_i')=n_i-k_i$. The central matrix-algebra conjecture. The following are equivalent:

• $\mathcal{C}_{A_1}^{\mathcal{C}}$ and $\mathcal{C}_{A_2}^{\mathcal{C}}$ are isomorphic as non-graded algebras;
• $\mathcal{C}_{A_1}^{\mathcal{C}}$ and $\mathcal{C}_{A_2}^{\mathcal{C}}$ are isomorphic as graded algebras;
• $A_1'$ and $A_2'$ are $z$-equivalent.

This extends the graph case to arbitrary matrices and is posed as an analogue of the proved classification for external zonotopal algebras. The source gives no resolution of the conjecture.