Holomorphic planes in M× T² from weakly exact Lagrangians in M

Let (M,ωM) be closed symplectic manifold containing a closed Lagrangian submanifold L with ωM vanishing on π₂(M,L). We prove that there is a nonconstant J-holomorphic plane u:C→ M× T² for any almost complex structure J on M× T² tamed by ωM⊕ωstd.