The PPT squared conjecture via decomposability of compositions

Let $\mathcal{M}_{d_i}$ denote the algebra of complex $d_i\times d_i$ matrices. Let $T:\mathcal{M}_{d_1}\rightarrow\mathcal{M}_{d_2}$ be completely positive and completely copositive, and let $P:\mathcal{M}_{d_2}\rightarrow\mathcal{M}_{d_3}$ be positive. A positive map $R:\mathcal{M}_{d_1}\rightarrow\mathcal{M}_{d_3}$ is decomposable if there are completely positive maps $R_1,R_2$ such that

$$R=R_1+\vartheta_{d_3}\circ R_2,$$

where $\vartheta_{d_3}$ is transposition on $\mathcal{M}_{d_3}$. PPT squared conjecture. The composition $P\circ T$ is decomposable. This is obtained in the paper as an equivalent formulation using duality of cones; the source gives no resolution, so it remains open.