Unique-limit conjecture for the smooth optimal transport gradient flow

Let $\rho_1$ be a log-concave density, meaning that $\log\circ\rho_1$ is concave. Consider the lifted gradient flow in the set of strictly convex functions, given by the flow referenced in the source.

Unique-limit conjecture. The flow has a unique limit in the set of strictly convex functions.

The conjecture is motivated by calculations showing that the Hessian of the relevant functional is non-positive when $\rho_1$ is log-concave. The source does not state whether the required infinite-dimensional convergence and uniqueness have been proved.