Dimension-free Lipschitz conjecture for Brenier maps onto log-Lipschitz Gaussian perturbations

Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^d$, and let $\mu$ be a log-Lipschitz perturbation of $\gamma$. Let $T=\nabla\varphi$ be the Brenier map transporting $\gamma$ onto $\mu$. Dimension-free Lipschitz conjecture. The map $T$ is globally Lipschitz, with Lipschitz norm bounded independently of the dimension. Even the case $\mu=\gamma$ is open at the time of the source, so the conjecture remains unresolved.