Instability of the ray-monotone selector for $W_1$-optimal transport

For the distance cost $c(x,y)=|x-y|$, the set $O(\mu,\nu)$ of $W_1$-optimal plans is generally not a singleton. Under the classical absolute-continuity hypotheses in the Euclidean case, secondary variational selection by the quadratic energy $C_2$ yields the ray-monotone $W_1$-optimal plan. We provide a counterexample to an open problem posed by Santambrogio that concerns the stability of this selector under weak convergence of the marginals. More precisely, we construct a fixed absolutely continuous source $\mu$ and absolutely continuous targets $\nu_n\rightharpoonup\nu$ such that $\gamma^{\mathrm{sel}}(\mu,\nu_n)\rightharpoonup\gamma^{\mathrm{hom}}$, where $\gamma^{\mathrm{hom}}\in O(\mu,\nu)$ but $\gamma^{\mathrm{hom}}\neq\gamma^{\mathrm{sel}}(\mu,\nu)$. We also identify the narrow Kuratowski limit of the optimal-plan sets $O(\mu,\nu_n)$, derive the constrained $\Gamma$-limit for secondary energies of the form $\int \Phi(|x-y|)\,d\gamma$ with $\Phi\in C([0,2])$, and deduce a non-commutation result for the additive perturbation $c_\varepsilon(x,y)=|x-y|+\varepsilon|x-y|^2$.