Barrett–Butler–Hall conjecture on equimodular inverses of symmetric 0–1 matrices

Let $n\geq 3$ and let $A\in\{0,1\}^{n\times n}$ be a nonsingular symmetric 0–1 matrix. The enhanced principal rank characteristic sequence of a symmetric matrix records, for each order, whether all, some but not all, or none of its principal minors are nonzero; saying that $A$ ends with $\tt{NAA}$ means that its last three entries are $\tt{NAA}$. A matrix is equimodular if all its entries have the same modulus, and a matrix has constant diagonal if all diagonal entries are equal. Barrett–Butler–Hall conjecture. The following statements are equivalent: (a) $A$ ends with $\tt{NAA}$; (b) $A^{-1}$ is equimodular with constant diagonal.

Equivalently, condition (a) says that all principal minors of $A$ of order $n-2$ are zero. One direction, namely (b) implies (a), has been established, but the converse remains unresolved.