  We consider the Dyson Ferrari-Spohn diffusion $\mathcal{X}^N =
(\mathcal{X}^N_1,\dots,\mathcal{X}^N_N)$, consisting of $N$ non-intersecting
Ferrari-Spohn diffusions $\mathcal{X}^N_1 > \cdots > \mathcal{X}^N_N > 0$ on
$\mathbb{R}$. This object was introduced by Ioffe, Velenik, and Wachtel (2018)
as a scaling limit for line ensembles of $N$ non-intersecting random walks
above a hard wall with area tilts, which model certain three-dimensional
interfaces in statistical physics. It was shown by Ferrari and Shlosman (2023)
that as $N\to\infty$, after a spatial shift of order $N^{2/3}$ and constant
rescaling in time, the top curve $\mathcal{X}^N_1$ converges to the
$\mathrm{Airy}_2$ process in the sense of finite-dimensional distributions. We
extend this result by showing that the full ensemble $\mathcal{X}^N$ converges
with the same shift and time scaling to the Airy line ensemble in the topology
of uniform convergence on compact sets. In our argument we formulate a Brownian
Gibbs property with area tilts for $\mathcal{X}^N$, which we show is equivalent
after a global parabolic shift to the usual Brownian Gibbs property introduced
by Corwin and Hammond (2014).
