First passage percolation on the exponential of two-dimensional branching random walk

We consider the branching random walk {R-z(N) : z is an element of 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(z)(gamma R)(N) 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 (N1-gamma 2/10).