  Long linear polymers in a depinned interfaces environment have been studied
for a long time, for instance in \cite{Caravenna2009depinning} when the
temperature is constant. In this paper, we study an extension of this model by
making the temperature and the distance between interfaces goes to infinity. We
give a full phase diagram for this model, and show that a new behaviour appears
in this extension. Our key tool include new sharp results on the simple random
walk evolving between interfaces.
