There is a well-known theorem stating that there is a bijection between diffeomorphism classes of Lefschetz fibrations over $S^2$ whose general fiber is a closed orientable surface $\Sigma_g$ of genus $g\geq 2$ and factorizations of identity in the mapping class group $\Gamma(\Sigma_g)$ up to Hurwitz moves and global conjugation. It is explained here, for example.
Is there an analogue of this theorem for symplectic Lefschetz fibrations in higher dimensions? In other words, are symplectomorphism classes of symplectic Lefschetz fibrations in bijective correspondence with factorizations of identity in the symplectic mapping class group of the general fiber?