  A symplectic manifold is called symplectic rationally connected if there is a
non-zero genus zero Gromov-Witten invariant with two point insertions. It is
conjectured that every smooth projective rationally connected variety is
symplectic rationally connected. In this short note we give some examples of
rationally connected 4-folds which are symplectic rationally connected. In
particular, all Fano 4-folds of pseudo-index at least 2 are symplectic
rationally connected. The proof does not use any explicit description of these
varieties.
