curve is balanced. Given the results in [@ShenMM], it is reasonable to expect that they are actually balanced. It is also easy to see that when $-K_X \cdot C=4$, the normal bundle of the very free curve is indeed balanced. If $-K_X \cdot C=5$, the curve $C$ is very free with balanced normal bundle ${{\mathcal}{O}}(1) \oplus {{\mathcal}{O}}(1) \oplus {{\mathcal}{O}}(1)$.

The $b_2(X)=2$ case is different in that we are unable to prove the existence of a non-zero Gromov-Witten invariant with two point insertions. Instead we prove a more precise result about uniruled $4$-fold whose second Betti number is $2$.

\[thm:KollarConj\] Let $W$ and $W'$ be two smooth projective uniruled $4$-fold which are symplectic deformation equivalent and has second Betti number $2$. Then the MRC fibration of $W$ and $W'$ has the same dimension and the class of a general fiber has the same cohomology class.

The idea of the proof is motivated by symplectic birational geometry (c.f. [@HLR]) which is the analogue of birational geometry in the symplectic category. This program in particular predicts that symplectic deformation equivalent varieties should have similar behaviors in birational geometry. One interesting question is whether the minimal model program (MMP) for a variety is something 