  We study the line ensembles of non-crossing Brownian bridges above a hard
wall, each tilted by the area of the region below it with geometrically growing
pre-factors. This model, which mimics the level lines of the $(2+1)$D SOS model
above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and
Wachtel. In those works, the tightness of the law of the top $k$ paths, for any
fixed $k$, was established under either zero or free boundary conditions, which
in the former setting implied the existence of a limit via a monotonicity
argument. Here we address the open problem of a limit under free boundary
conditions: we prove that as the interval length, followed by the number of
paths, go to $\infty$, the top $k$ paths converge to the same limit as in the
free boundary case, as conjectured by Caputo, Ioffe and Wachtel.
