  We are concerned with the half-space Dirichlet problem
\[\left\{\begin{array}{ll}
  -\Delta v+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+,
  v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\
\textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array}\right. \] where
$\mathbb{R}^N_+=\{x\in \mathbb{R}^N \ : \ x_N>0\}$ for some $N\geq 2$, and
$p>1$, $c>0$ are constants. It was shown recently by Fernandez and Weth [Math.
Ann. (2021)] that there exists an explicit number $c_p\in (1,\sqrt{e})$,
depending only on $p$, such that for $0<c<c_p$ there are infinitely many
bounded positive solutions, whereas, for $c>c_p$ there are no bounded positive
solutions. They also posed as an interesting open question whether the
one-dimensional solution is the unique bounded positive solution in the case
where $c = c_p$. If $N=2, 3$, we recently showed this one-dimensional symmetry
property in [Partial Differ. Equ. Appl. (2021)] by adapting some ideas from the
proof of De Giorgi's conjecture in low dimensions. Here, we first focus on the
case $1<p<3$ and prove this uniqueness property in dimensions $2\leq N\leq 5$.
Then, for the cubic NLS, where $p = 3$, we establish this for $2 \leq N \leq
4$. Our approach is completely different and relies on showing that a suitable
auxiliary function, inspired by a Lyapunov-Schmidt type decomposition of the
solution, is a nonnegative super-solution to a Lane-Emden-Fowler equation in
$\mathbb{R}^{N-1}$, for which an optimal Liouville type result is available.
