Undecidability of tensor stable positive maps

Let $d\in\mathbb{N}$ and let $\mathcal{P}:\mathcal{M}_d\to\mathcal{M}_d$ be a linear map whose Choi matrix has entries in $\mathbb{Q}$. The problem tensor stable positivity asks whether

$$\mathcal{P}^{\otimes n}\text{ is positive for all }n.$$

Tensor stable positivity conjecture. The set of yes-instances of tensor stable positivity, denoted $\textsc{tsp}$, is not recursively enumerable.

The problem is already known to be co-recursively enumerable, since a no-instance can be certified by finding an $n$ for which $\mathcal{P}^{\otimes n}$ is not positive. The conjecture asserts that yes-instances cannot likewise be recognised by a Turing machine, and hence that tensor stable positivity is undecidable.