Exact earth mover's correlation formula for bivariate normal variables

Let $(X,Y)$ be bivariate normal random variables, and write $\varrho(X,Y)=\varrho$ for their correlation. The bivariate normal earth mover's correlation conjecture. One has

$$\mathop{\mathrm{eCor}}(X,Y)=\left[1-\sqrt{1-\varrho^2}\,\right]^{1/2}.$$

The preceding theorem proves the corresponding upper bound, while this conjecture asserts that the bound is attained. Its resolution is not specified in the supplied text.