Existence conjecture for connected contactifications of compact symplectic manifolds

Let $(N,\omega)$ be a compact symplectic manifold, and let $\hbar>0$. The Dirac quantization condition is

$$\big[\omega/2\pi\hbar\big]\in H^2(N,\mathbb{Z}).$$

A contactification of $(N,\omega)$ is a contact-geometric lift as defined in the source, and it is connected when its total space is connected. Contactification existence conjecture. The manifold $(N,\omega)$ admits a connected contactification if and only if it satisfies the Dirac quantization condition for some $\hbar>0$.

The conjecture identifies the Dirac quantization condition as the exact obstruction to constructing a connected contactification of a compact symplectic manifold. The source states that this existence question remains open, since contactifications can be topologically complicated.