Georgiou–Pavon conjecture for positive quantum channels

Let $Q:\mathbb{C}^{n\times n}\to\mathbb{C}^{n\times n}$ be a positive quantum channel, and let $\rho_0,\rho_T$ be density matrices. Let $\mathrm{H}_{n,++}$ denote the positive definite Hermitian $n\times n$ matrices and let $\mathrm{H}_{n,+}$ denote the positive semidefinite Hermitian $n\times n$ matrices. Georgiou–Pavon conjecture. There exist $\phi_0,\phi_T,\hat\phi_0,\hat\phi_T\in\mathrm{H}_{n,++}$ such that

$$Q(\phi_T)=\phi_0,\qquad Q'(\hat\phi_0)=\hat\phi_T,$$ $$\rho_0=\chi_0\hat\phi_0\chi_0^*,\qquad \rho_T=\chi_T\hat\phi_T\chi_T^*,$$ $$\phi_0=\chi_0^*\chi_0,\qquad \phi_T=\chi_T^*\chi_T.$$

Furthermore, $\chi_0$ and $\chi_T$ can be chosen in $\mathrm{H}_{n,+}$. The paper states that its Theorem proves this conjecture for two positive definite density matrices, so the claim is solved in that setting; the formulation above is the conjecture attributed to Georgiou and Pavon.