Gompf's embedding conjecture for closed oriented 3-manifolds

Let $Y$ be a closed oriented $3$--manifold. Gompf's embedding conjecture. $Y$ admits a smooth embedding into a symplectic $4$--manifold.

Such an embedding would provide a route toward constructing simply connected $4$--manifolds with infinitely many smooth structures bounding arbitrary closed oriented $3$--manifolds. The source presents this as an interesting question and gives no resolution.