The PPT-squared conjecture for entanglement-breaking quantum channels

Let $\matrices$ denote the matrix algebra in the paper. A linear map $\psi:\matrices\to\matrices$ is PPT if it is completely positive and completely copositive, meaning that both $\psi$ and $\operatorname{(\cdot)^T}\circ\psi$ are completely positive, where transpose is taken with respect to the computational basis of $\mathbb{C}^d$. A map is entanglement breaking if, for every $k\in\mathbb{N}$ and every positive semidefinite $\rho\in\matrices\otimes \mathbb{M}_k$, the matrix $(\psi\otimes\operatorname{Id}_{\mathbb{M}_k})(\rho)$ is separable. $\mathrm{PPT}^2$ conjecture. If $\psi,\varphi:\matrices\to\matrices$ are PPT, then $\psi\circ\varphi$ is entanglement breaking. This conjecture concerns whether the composition of any two PPT maps necessarily destroys entanglement, and is a central question in the theory of entanglement-breaking channels.