Christandl's PPT$^2$ conjecture for entanglement-breaking compositions

Let $\Phi_1,\Phi_2\in\mathcal{T}_n(\mathbb{C})$ be linear maps. A map $\Phi$ is PPT when both $\Phi$ and $\Phi\circ\mathsf{T}$ are completely positive, and it is entanglement breaking when $[I_n\otimes\Phi](Y)\in\operatorname{SEP}_n$ for every $Y\in\operatorname{Herm}(\mathbb{C}^n\otimes\mathbb{C}^n)_+$. PPT$^2$ conjecture. The composition $\Phi_1\circ\Phi_2$ of two arbitrary PPT linear maps $\Phi_1,\Phi_2$ is entanglement breaking. Equivalently, if $\rho_1,\rho_2\in\operatorname{Herm}(\mathbb{C}^n\otimes\mathbb{C}^n)$ are the corresponding Choi states and $\rho_1,\rho_2\in\operatorname{DPS}^{(1)}_n$, then

$$J(\Phi_{\rho_1}\circ\Phi_{\rho_2})\in\operatorname{SEP}_n.$$

Christandl posed this conjecture in 2012; it asks whether composing two PPT maps always destroys entanglement, and its resolution remains open.