The optimal coupling conjecture for earth mover's covariance

Let $X$ and $Y$ be random variables in metric spaces with metric $\delta$, and let $(X',Y')$ range over admissible couplings having the prescribed marginal laws. Define the earth mover's covariance by

$$\mathop{\mathrm{eCov}}(X,Y)=\inf_{(X',Y')}\mathbb E\big[\delta(X,X')+\delta(Y,Y')\big].$$

The optimal coupling conjecture. Under “general conditions”, the infimum can be attained or computed among couplings satisfying $X=X'$ or $Y=Y'$. The text presents this as a conjectural simplification of the optimization problem, but does not specify which general conditions suffice.