Second-level collapse conjecture for the PPT entanglement-cost hierarchies

Let $\rho=\rho_{AB}$ be a state. The quantities $E_{\chi,p}(\rho)$ are the $\chi$-quantities defined by the hierarchy at level $p$, and $E_{\kappa,p}(\rho)$ are the corresponding $\kappa$-quantities. Write $E^{\mathrm{exact}}_{c,\mathrm{PPT}}(\rho)$ for the exact entanglement cost under PPT operations. Second-level collapse conjecture. For every state $\rho=\rho_{AB}$,

$$E_{\chi,3}(\rho)=E_{\chi,2}(\rho).$$

Consequently, the two hierarchies collapse and

$$E^{\mathrm{exact}}_{c,\mathrm{PPT}}(\rho)=E_{\chi,2}(\rho)=E_{\kappa,3}(\rho).$$

The conjecture concerns whether the hierarchy limit can be computed at a finite level. The paper reports numerical evidence for collapse at the second level of the $\chi$-hierarchy, but states that it was neither confirmed nor disproved analytically.