  We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with
Gaussian increments indexed over a two-dimensional box $V_N$ of side length
$N$, and we study the first passage percolation where each vertex is assigned
weight $e^{\gamma \mathcal R^N_z}$ for $\gamma>0$. We show that for $\gamma>0$
sufficiently small but fixed, the expected FPP distance between the left and
right boundaries is at most $O(N^{1 - \gamma^2/10})$.
