Undecidability conjecture for ALLOWS_QUANTUM

Let $H$ be a contextuality scenario and let $C^*(H)$ denote its universal unital $C^*$-algebra. The decision problem $\mathtt{ALLOWS\_QUANTUM}$ asks whether a given contextuality scenario admits a quantum realization. Undecidability conjecture. The decision problem $\mathtt{ALLOWS\_QUANTUM}$ is undecidable. The authors do not know any terminating algorithm that solves this problem; a positive solution to the associated residual finite-dimensionality problem would instead yield an algorithm, while the problem is related to Connes' embedding problem and other difficult questions in operator algebras.