The sparsity conjecture for minimally AP-irreducible sign pattern matrices

Let $A$ be a minimally AP-irreducible sign pattern matrix of order $n$ whose diagonal entries are all zero. A sign pattern matrix $A$ is minimally AP-irreducible if it is AP-irreducible and deleting any nonzero entry destroys AP-irreducibility. Sparsity conjecture. The matrix $A$ has at most $2n-2$ nonzero entries.

The paper establishes this bound under the additional hypothesis that $A[\alpha,\beta]$ contains no $+$ for any two distinct irreducible components $\alpha$ and $\beta$ of $A_+$. The conjecture asks whether the same bound holds without that additional hypothesis; its status is not resolved in the supplied text.