The IPT-squared conjecture for real and complex matrix algebras

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$, and let $\Phi:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$ be an IPT map, meaning that its Choi matrix is invariant under partial transposition. A map is $\mathbb{K}$-entanglement breaking when it breaks entanglement over the field $\mathbb{K}$. IPT-squared conjecture. Every IPT map satisfies that $\Phi\circ\Phi$ is $\mathbb{K}$-entanglement breaking.

This conjecture is a field-relative version of the PPT-squared conjecture. The direct real-vector-space analogue using PPT maps is false, even for $n=2$, whereas the IPT formulation is proposed for both $\mathbb{R}$ and $\mathbb{C}$; its general status remains open.