Polyhedral eigenvalue conjecture for regular matroid representations

Let $\mathcal{M}$ be a regular matroid and let $M$ be a unimodular $m \times n$ representation of $\mathcal{M}$ with corank greater than $1$. Let $L$ denote the associated matrix whose nonzero eigenvalues are under consideration. Polyhedral eigenvalue conjecture. There is an $m \times m$ matrix $\Lambda$ with full rank such that every nonzero eigenvalue of $L$ is an eigenvalue of $\Lambda$, and every other eigenvalue of $\Lambda$ depends only on the ambient dimension $m$. The conjecture proposes an extension of the preceding corank-one construction to unimodular representations of regular matroids of arbitrary corank; no proof is given in the source.