Ballisticity for positive excited random walks without bounded cookie stacks

We consider a nearest-neighbour excited random walk on the integers in an iid cookie environment. Every cookie has nonnegative drift, and the environment satisfies weak ellipticity. If $\delta$ is the expected total drift at one site, then the limiting speed is zero for $\delta\le2$ and strictly positive for $\delta>2$. No bound on the number or positions of the non-neutral cookies, and no decay assumption on the stack depth, is required. This answers Problem 5.3 of Kosygina and Zerner. The proof at $\delta=2$ represents each cookie as a mixture of a deterministic right step and a fair step, then moves the deterministic steps to the front of the stack. The resulting branching chain has arbitrary integrable mean-two emigration, with positive probability of no emigration. A convex-order comparison with a three-point emigration law shows that its stationary mean is infinite. For the three-point law, this follows from a generating-function identity. A direct downcrossing argument then gives zero speed.