Boundedness of semilinear Duffing equations with Liouvillean frequency

We are concerned with the quasi-periodic semilinear Duffing equation $ x''+\omega^2x+g(x,t) = 0, $ where $ \omega $ is a Diophantine number, $ g(x,t) $ is bounded, real analytic in $ x $ and $ t $, and is quasi-periodic in $ t $ with the frequency $ \tilde{\omega} = (1, \alpha) $, where $ \alpha $ is Liouvillean. Without assuming the twist condition and the polynomial-like condition on this equation, we will prove the boundedness of all solutions.