Catalano–Lavenant farthest-coupling conjecture

Let $\lambda$ denote the uniform probability measure on $[0,1]$, let $\Pi(\lambda,\lambda)$ be the set of Borel probability measures on $[0,1]^2$ whose two marginals are $\lambda$, and let $\lambda\otimes\lambda$ be the independent coupling. Define the monotone and antimonotone couplings by $\pi_{+}:=(t\mapsto(t,t))_{\#}\lambda$ and $\pi_{-}:=(t\mapsto(t,1-t))_{\#}\lambda$. Then $\pi_{+}$ and $\pi_{-}$ maximize the Wasserstein distance from the independent coupling: $W_2(\pi_{+},\lambda\otimes\lambda)=W_2(\pi_{-},\lambda\otimes\lambda)=\sup_{\pi\in\Pi(\lambda,\lambda)}W_2(\pi,\lambda\otimes\lambda)$, where the Wasserstein distance is computed on $[0,1]^2$ with its Euclidean metric.