Positive–PPT composition decomposability conjecture

A linear map is positive if it maps positive semidefinite operators to positive semidefinite operators, and it is PPT if it is completely positive and has a PPT Choi matrix. Let $\mathcal{L}_1$ be a positive linear map and let $\mathcal{L}_2$ be a PPT linear map.

Positive–PPT composition conjecture. The composition

$$\mathcal{L}_1\circ\mathcal{L}_2$$

is decomposable. This is presented as an equivalent formulation of the PPT squared conjecture; since the latter is resolved only for $d=2$ and $d=3$, this formulation remains open in general.