Summary: 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 \(O(N^{1 - \gamma^2/10})\).