  We consider the 3d cubic nonlinear Schr\"odinger equation (NLS) with a strong
2d harmonic potential. The model is physically relevant to observe the
lower-dimensional dynamics of the Bose-Einstein condensate, but its ground
state cannot be constructed by the standard method due to its supercritical
nature. In Bellazzini-Boussa\"id-Jeanjean-Visciglia \cite{BBJV}, the authors
constructed a proper ground state introducing a constrained energy minimization
problem. In this paper, we further investigate the properties of the ground
state. First, we show that as the partial confinement is increased, the 1d
ground state is derived from the 3d energy minimizer with a precise rate of
convergence. Then, by employing this dimension reduction limit, we prove the
uniqueness of the 3d minimizer provided that the confinement is sufficiently
strong. Consequently, we obtain the orbital stability of the minimizer, which
improves that of the set of minimizers in the previous work \cite{BBJV}.
