Strong fillability implies nonvanishing contact homology

Let $(M,\xi=\ker \alpha)$ be a strongly fillable contact manifold, and let $HC_*(M,\xi)$ denote its full contact homology.

Nonvanishing conjecture.

$$HC_*(M,\xi)\neq 0.$$

If true, this would give a contact-homological obstruction to strong fillability for contact manifolds with vanishing contact homology. The statement is presented as a well-known observation, but no resolution is supplied here.