Positive semidefinite matrix characterization of the range of Ψ

Prove whether every checkerboard copula A whose value matrix S_A, mass matrix M_A, and density matrix G_A are positive semidefinite can be represented as A=Ψ(C) for some bivariate copula C.

Background

The paper proves that if a checkerboard copula A belongs to the range of Ψ, then its value, mass, and density matrices are positive semidefinite. The converse implication is displayed as (P4) in the schematic summary: positive semidefiniteness of all three associated matrices would imply that A is Ψ(C) for some general copula C.

This problem concerns whether the positive-semidefinite matrix conditions provide a complete characterization of checkerboard copulas in the range of Ψ. The authors note that if (P3) were true, then (P4) would be false because there exist entrywise nonnegative positive-semidefinite matrices that are not completely positive.

References

We collect the findings of the last three theorems in a schematic summary, along with two implications that are still open, see Figure~\ref{fig:results}.

— On diagonal sections of Markov products of copulas  (2610.13846 - Drnovšek et al., 9 Oct 2026) in Section 3, Figure 1 (implication (P4)); concluding remarks, Section 7