Second-smallest support size in three-dimensional polystochastic matrices

Determine the next minimum value of the support size among three-dimensional polystochastic matrices of order n after the minimum attained by multidimensional permutations.

Background

The paper notes that multidimensional permutations attain the minimum possible support size among matrices in the polytopes of polystochastic matrices. Fischer and Swart posed the problem of identifying the next larger possible support size for matrices in the three-dimensional case. The paper explicitly states that this question remains unresolved and does not solve it.

References

Fischer and Swart asked what the next minimum value for the support of matrices in $\Omega_n3$ is, but this question remains unanswered.

— Enumeration and constructions of vertices of the polytope of polystochastic matrices  (2502.09149 - Taranenko, 13 Feb 2025) in Section 1, Introduction