Stone’s conjecture for fully semimonotone Q0-matrices

Prove that every fully semimonotone Q0-matrix is a P0-matrix, equivalently, establish the inclusion E0^f ∩ Q0 ⊆ P0 for matrices of arbitrary order, including the singular case.

Background

Stone proved the analogous inclusion U ∩ Q0 ⊆ P0, but his proof did not extend to the larger class E0f of fully semimonotone matrices. The conjecture asks whether membership in both E0f and Q0 guarantees nonnegative principal minors. Earlier work verified the claim in low dimensions, under additional hypotheses, and for several special subclasses, but did not establish it in general.

The paper proves the claim only under the additional hypothesis det A > 0. Its example shows that positivity of the determinant cannot be inferred from Q0-membership. Consequently, the unresolved portion includes matrices in E0f ∩ Q0 with det A = 0; the authors state that this singular case is not addressed by their induction.

References

Stone observed that his proof of Theorem~\ref{thm:stone-UQ0} does not extend to the larger class $E$, and left the following as an open question: \begin{conjecture}[Stone] \label{conj:stone} Every $E \cap Q$-matrix is a $P$-matrix. \end{conjecture}

— Partial Progress on Stone's Conjecture: $P_0$-Membership of Fully Semimonotone Matrices with Positive Determinant  (2608.22829 - Ghosh, 24 Aug 2026) in Section 1, Introduction; Conjecture 1 (labeled conj:stone); Section 5, Conclusion