Mesa limit and uniform projection conjecture

Let $\mathcal U$ denote the class of compactly supported uniform measures

$$\mathcal U=\left\{\operatorname{Vol}(A)^{-1}\mathcal L^D|_A:\ A\subset\mathbb{R}^D\text{ compact},\ 0<\operatorname{Vol}(A)<\infty\right\}.$$

After replacing the density-ratio objective by a density-cap constraint, consider the Wasserstein gradient flows of the Rényi entropies $\mathcal E_m$ as $m\to\infty$.

Mesa limit and uniform projection conjecture. The flows should converge to a Hele–Shaw or mesa-type free-boundary evolution whose terminal state is the Wasserstein projection of $\mu$ onto $\mathcal U$.

Classical convergence results and free-boundary theory support this picture, but the full projection statement has not been proved in the present sampleability setting.