The flow map of the Fokker-Planck equation does not provide optimal transport

In Khrulkov and Oseledets (2022) the authors conjecture that, by integrating the flow of the ODE given by the Wasserstein velocity in a Fokker-Planck equation, one obtains an optimal transport map. On the other hand this result was thought to be false in Kim and Milman (2012) but no proof was provided. In this note we show that the result claimed by Khrulkov and Oseledets cannot hold. This strengthens a counterexample which was built in Tanana (2021). (c) 2022 Elsevier Ltd. All rights reserved.