The PPT squared conjecture via tensor products of PPT maps

Let $\mathcal{M}_{d_i}$ denote the algebra of complex $d_i\times d_i$ matrices. Let

$$T_1:\mathcal{M}_{d_1}\rightarrow\mathcal{M}_{d_2},\qquad T_2:\mathcal{M}_{d_3}\rightarrow\mathcal{M}_{d_4}$$

be linear maps, each completely positive and completely copositive. For a positive matrix $X\in(\mathcal{M}_{d_1}\otimes\mathcal{M}_{d_3})^+$, write $(T_1\otimes T_2)(X)$ for its image. A matrix is separable if it is a sum of positive product matrices. PPT squared conjecture. The matrix $(T_1\otimes T_2)(X)$ is separable for every positive matrix $X\in(\mathcal{M}_{d_1}\otimes\mathcal{M}_{d_3})^+$. This is another reformulation of the PPT squared conjecture, connected to locally entanglement-annihilating maps; the source supplies no resolution, so the claim remains open.