The optimal transport problem can admit many solutions, for instance when the cost function is the Euclidean distance on R^d (Monge's problem). On the other hand, entropically regularized optimal transport (EOT) always has a unique optimizer. It is a longstanding open problem whether EOT selects some particular optimal transport coupling in the vanishing regularization limit, and what that coupling would look like. We provide a surprising answer for Monge's problem in dimension d > 1. (Joint work with Chenyang Zhong.)