Thanks Henry! I am convinced that the answer to my question is unknown.

Thanks! Are there cases of $3$-manifolds (with infinite $\pi_1)$) for which we know Simple Loop Conjecture holds? Would it be helpful if we assume that the group $G$ has three (or less) generators and abelianization of rank 0.

Thank you. Can I apply Lefschetz hyperplane theorem if it is not known $L$ can be represented by a complex submanifold? All I know is my $\Sigma_k$ is a symplectic submanifold of $X$.

Maybe they admit complex structure. not sure about that. Probably this seems very difficult question.

I was just checking the following papers on arxiv. They do contain some symplectic, but non Kahler Calabi-Yau 6 manifolds. Here are the links arxiv.org/pdf/1107.2623.pdf arxiv.org/abs/1105.3519