Catalano–Lavenant conjecture on extremal uniform couplings

Establish that, among all couplings of two uniform probability measures on [0,1], the diagonal coupling and the antidiagonal coupling maximize the 2-Wasserstein distance from the independent coupling.

Background

The paper studies couplings with prescribed one-dimensional marginals that are farthest, in 2-Wasserstein distance, from the independent coupling. For uniform marginals on [0,1], such couplings are precisely bivariate copulas, and the monotone and antimonotone couplings are supported on the diagonal and antidiagonal of the unit square, respectively.

Catalano and Lavenant conjectured that these two extreme dependence structures maximize the Wasserstein distance from independence. The paper proves an analogous result for arbitrary numbers of one-dimensional standard Gaussian marginals, but does not resolve the uniform-marginal conjecture itself.

References

This led Catalano and Lavenant to formulate the following conjecture, which remains open to the best of our knowledge Remark~3.

\begin{conjecture}[{Remark~3] Let $\mu$ denote the uniform probability measure on $[0,1]$. Then the diagonal and antidiagonal couplings $(\mathrm{id},\mathrm{id})#\mu$ and $(\mathrm{id},1-\mathrm{id})#\mu$ maximize the 2-Wasserstein distance from the independent coupling $\mu\otimes\mu$ among all elements of $\Pi(\mu,\mu)$. \end{conjecture}

— Couplings Farthest from the Independent Gaussian  (2609.10467 - Schrott, 9 Sep 2026) in Section 1, Introduction, paragraph preceding the conjecture