Local distinguishability of nested block-positive cones in non-generic cases

Prove that the inclusion of convex sets of k-blockpositive matrices BPℓ ⊂ BPℓ+1 is locally distinguishable around the projection state ϱ_{E^⊥} whenever ℓ ≥ k and ϱ_{E^⊥} lies on the boundary of BP_k, including non-generic subspaces E.

Background

The paper studies the local geometry of the convex set BP_k of k-blockpositive matrices near a projection state ϱ{E^⊥}, where E^⊥ is the orthogonal-complement subspace associated with the face of states whose ranges are contained in E. Schmidt number witnesses are related to the differences BP{ℓ−1} \ BP_ℓ, and local distinguishability of the inclusions BP_ℓ ⊂ BP_{ℓ+1} can be detected through supporting hyperplanes.

For generic subspaces E, the authors establish the existence of Schmidt number witnesses around the relevant projection states using algebraic-geometric information about vectors in E of bounded Schmidt rank. They explicitly state that their method does not establish this existence for non-generic cases. They then conjecture that the corresponding inclusions remain locally distinguishable whenever the projection state lies on the boundary of BP_k, suggesting that nonlinear boundary geometry may be required to resolve the question.