Stone’s conjecture on fully semimonotone Q₀-matrices

Stone conjectured that every fully semimonotone $Q_0$-matrix has nonnegative principal minors, equivalently belongs to $P_0$. The conjecture was introduced by Stone and was still open in the 2013 literature.

Known results

• Murthy and Parthasarathy proved the result for fully copositive $Q_0$-matrices.
• A 2013 paper established it when the complementary cones have no partial incidence.
• Cao and Ferris proved the corresponding equality for matrices of order two.
• Later work treated subclasses of semimonotone-star matrices, but did not settle the full conjecture.

August 2026 determinant-positive case

A new preprint claims, for arbitrary order, that $E_0^f\cap\{A:\det A>0\}\subseteq P_0$. It also constructs a fully semimonotone matrix with positive determinant outside $Q_0$, showing that positive determinant alone cannot prove Stone’s conjecture. These claims are unverified and leave the original $Q_0$ case open.

Current status (as of August 2026): Stone’s conjecture remains open; a preprint claims substantial determinant-positive progress and a limiting counterexample, but neither has been independently verified.