Letrouit's square-root stability conjecture for Brenier maps from a uniform convex source

Theorem 1.1: for $K$ compact convex with interior in $\mathbb R^d$, $d\ge2$, and $Y$ compact, $\|T_\mu-T_\nu\|_{L^2(\rho)}\le C(K,Y)W_2(\mu,\nu)^{1/3}$ for all probability measures $\mu,\nu$ on $Y$, atomic or not; the exponent $1/3$ cannot be raised even for $K=Y=[-1,1]^d$ and three-atom targets, so Letrouit's square-root conjecture is false and $1/3$ is the sharp uniform exponent. Not shown: sharp exponents in $W_1$ (a sharp $W_1^{1/4}$ bound is attributed to Merigot) or for nonuniform source densities.

Theorem 1.1: for $K$ compact convex with interior in $\mathbb R^d$, $d\ge2$, and $Y$ compact, $\|T_\mu-T_\nu\|_{L^2(\rho)}\le C(K,Y)W_2(\mu,\nu)^{1/3}$ for all probability measures $\mu,\nu$ on $Y$, atomic or not; the exponent $1/3$ cannot be raised even for $K=Y=[-1,1]^d$ and three-atom targets, so Letrouit's square-root conjecture is false and $1/3$ is the sharp uniform exponent. Not shown: sharp exponents in $W_1$ (a sharp $W_1^{1/4}$ bound is attributed to Merigot) or for nonuniform source densities.