  Let $\{Y_{\mathfrak{B}}(x)\,:\,x\in\mathfrak{B}\}$ be a discrete Gaussian
free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with
Dirichlet boundary conditions. We study Liouville first-passage percolation:
the shortest-path metric in which each vertex $x$ is given a weight of
$e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma>0$. We show that for
sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_{k}\}$
there exists a subsequence along which the appropriately scaled and
interpolated Liouville FPP metric converges in the Gromov--Hausdorff sense to a
random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible
(conjecturally unique) scaling limits are homeomorphic by
bi-H\"older-continuous homeomorphisms to the unit square with the Euclidean
metric.
