Random walks are determined by their trace on the positive half-line

We prove that the law of a random walk $X_n$ is determined by the one-dimensional distributions of $\max(X_n, 0)$ for $n = 1, 2, \ldots$, as conjectured recently by Lo\"ic Chaumont and Ron Doney. Equivalently, the law of $X_n$ is determined by its upward space-time Wiener-Hopf factor. Our methods are complex-analytic.