Determine the minimal dimension of a counterexample to the Foregger–Sinkhorn tie-point conjecture

Determine whether the dimension-eight counterexample to the Foregger–Sinkhorn tie-point conjecture is minimal by proving or disproving the existence of a counterexample in a smaller dimension.

Background

The paper constructs an order-eight nearly decomposable support and a unique global minimizer on its associated doubly stochastic face that violates the Foregger–Sinkhorn tie-point implication. Although this establishes that the conjecture is false, the result does not determine the smallest dimension in which such a counterexample can occur.

The unresolved issue is therefore the minimality of the dimension eight: a smaller counterexample may exist, but the paper does not prove that none does. Establishing minimality would complete the dimensional classification of counterexamples to the conjecture.