Contact deformations of closed 1-forms on T2-bundles over S1

If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. On any other 2-dimensional torus bundle over the circle, every closed, nonsingular 1-form deforms linearly into contact forms.