Gap between E_{χ,2} and E_{χ,3}

Ascertain whether there exists, in general, a gap between the second and third levels of the χ-hierarchy; specifically, determine whether there are bipartite states ρ for which E_{χ,2}(ρ) < E_{χ,3}(ρ), or else prove that E_{χ,2}(ρ) = E_{χ,3}(ρ) holds for all ρ.

Background

The χ-hierarchy generalizes logarithmic negativity and provides lower bounds that converge to the exact zero-error PPT entanglement cost. The authors found explicit examples where E_{χ,0}(ρ) < E_{χ,1}(ρ) and E_{χ,1}(ρ) < E_{χ,2}(ρ), motivating the question whether a further increase occurs between levels 2 and 3.

If E_{χ,2}(ρ) = E_{χ,3}(ρ) for all states, then the hierarchy would effectively collapse at level 2, yielding a single-letter formula for the cost.