The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States

For a bipartite state $ρ$, information about the spectrum of its partial transpose $ρ^{Γ_B}$ can be inferred from measurements on multiple copies of $ρ$, without full state tomography. This raises a natural question: which eigenvalue lists can arise as $\operatorname{spec}(ρ^{Γ_B})$ for a density op