  In this note, we generalise a result of Lalonde, McDuff and Polterovich
concerning the $ C^0 $ flux conjecture, thus confirming the conjecture in new
cases of a symplectic manifold. Also, we prove the continuity of the flux
homomorphism on the space of smooth symplectic isotopies endowed with the $ C^0
$ topology, which implies the $ C^0 $ rigidity of Hamiltonian paths,
conjectured by Seyfaddini.
