Most insight in the problem probably emerges when approaching the effect form Landau Theory. Which is a little bit problematic as the melting of a bulk in general has to be considered as a first order phase transition, meaning the order parameter  jumps at . The derivation of Lipowski (basic geometry shown in fig.2) leads to the following results when :
Where  is the order parameter at the border between (2) and (l),  the so-called extrapolation length and  a constant that enters the model and has to be determined using experiment and other models. Hence one can see that the order parameter in the liquid film can undergo a continuous phase transition for large enough extrapolation length. A further result is that  what corresponds to the result of the thermodynamical model in the case of short range interactions. 
Landau Theory does not consider fluctuations like capillary waves, this could change the results qualitatively.