It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma) $ where $s=s(\xi)$ is the spin$^c$-structure associated to $\xi$, $e(s)$ is its Euler class, and $\Sigma \subset M$ is an embedded surface, not the sphere.

The question is about a converse of this statement. Namely, if a spin$^c$ structure $s$ satisfies the above inequality for all surfaces $\neq S^2$, can we conclude that there is a tight contact structure representing $s$?

By the way, what can we say in general about such spin$^c$-structures?

P.s. For the sphere we must have $\langle e(s),[S^2] \rangle =0$.