Recently, attention has been given to a system of He$^3$ bi-layers where a quantum criticality similar to the one in heavy fermion compounds has been observed [Science 317, 1356 92007)] In our previous analysis [Phys. Rev. B 79, 045112 (2007)], based on the Kondo breakdown scenario, we addressed successfully most of the features observed in that experiment. Here, we consider the activation energy $\Delta$ observed experimentally in the specific heat measurements at low temperatures in the heavy Fermi liquid phase. Within our previous study of this system, this is identified with the gap opening when the upper hybridized band is emptied due to a strong hybridization between the nearly localized first layer and the fluid second one. We discuss the successes and limitations of our approach. An additional prediction is proposed.