We will investigate the effects of the lack of compactness in the critical Folland-Stein(-Sobolev) embedding in the Heisenberg group. In particular, by means of Γ-convergence techniques, we will show that optimal functions of a subcritical energy approximation of the Sobolev quotient in bounded (not necessarily regular) domains do concentrate energy at a single point. Moreover, in clear accordance with the underlying geometry, we show that the concentration of the energy happens at a critical point for the Robin function (i. e., the diagonal of the regular part of the Green function associated to the involved domain), thus proving that an analogous of a conjecture of Brezis and Peletier does still hold in the Heisenberg group.