  Let $\{\eta(v): v\in V_N\}$ be a discrete Gaussian free field in a
two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary
conditions. We study the Liouville first passage percolation, i.e., the
shortest path metric where each vertex is given a weight of $e^{\gamma
\eta(v)}$ for some $\gamma>0$. We show that for sufficiently small but fixed
$\gamma>0$, with probability tending to $1$ as $N\to \infty$, all geodesics
between vertices of macroscopic Euclidean distances simultaneously have (the
conjecturally unique) length exponent strictly larger than 1.
