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 $n=1,2,\dots$, 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.