The convex-fillability characterization of contact-type embeddability

Let $\Xi(3)$ denote the collection of closed $3$--manifolds with positive, cooriented contact structures. Let $\Xi_{\text{\textsf{embed}}}(3)$ consist of those $(M,\xi)\in\Xi(3)$ that admit a contact type embedding into some closed symplectic manifold, and let convex fillability mean the existence of a convex symplectic filling. Embeddability conjecture. $\Xi_{\text{\textsf{embed}}}(3)$ is the set of convex fillable contact $3$--manifolds; equivalently, if $(M,\xi)$ is not convex fillable, then it admits no contact type embedding into any closed symplectic manifold. The paper presents this as an open question motivated by filling obstructions from $J$--holomorphic curves; the claimed equivalence remains unresolved in the supplied text.