Surviving from the tip of a cone in competing first-passage percolation

In two-type first passage percolation on $\mathbb{Z}^2$, two entities compete to capture the sites of the lattice. The entities spread between nearest neighbor sites at times specified by random passage times associated with the edges. We consider the case when both types have the same passage time distribution, with one type starting at the origin and the other from an infinite cone with tip at the origin and pointing in direction $\theta$. Itai Benjamini has suggested that the type starting at the origin can grow unboundedly if and only if the slope of the cone is strictly smaller than $\pi/2$, so that the cone does not fill a whole half-plane. The main result is that this is correct for any $\theta$ such that the asymptotic shape of the one-type process has a tangent line with direction $\theta$. The proofs are based on a description of infinite time-minimizing paths in terms of Busemann functions together with local modification arguments.