pr.probability - Is the infimum of the Ky Fan metric achieved?

Consider the probability space $(Omega, {cal B}, lambda)$ where
$Omega=(0,1)$, ${cal B}$ is the Borel sets, and $lambda$ is Lebesgue measure.

For random variables $W,Z$ on this space, we define the Ky Fan metric by

$$alpha(W,Z) = inf lbrace epsilon > 0: lambda(|W-Z| geq epsilon) leq epsilonrbrace.$$

Convergence in this metric coincides with convergence in probability.

Fix the random variable $X(omega)=omega$, so the law of $X$ is Lebesgue measure,
that is, ${cal L}(X)=lambda$.

Question: For any probability measure $mu$ on $mathbb R$, does there exist
a random variable $Y$ on $(Omega, {cal B}, lambda)$ with law $mu$ so that
$alpha(X,Y) = inf lbrace alpha(X,Z) : {cal L}(Z) = murbrace$ ?

Notes:

1. By Lemma 3.2 of Cortissoz,
   the infimum above is $d_P(lambda,mu)$:
   the Lévy-Prohorov distance between the two laws.




2. The infimum is achieved if we allowed to choose both random variables.
   That is, there exist $X_1$ and $Y_1$ on $(Omega, {cal B}, lambda)$
   with ${cal L}(X_1) = lambda$, ${cal L}(Y_1) = mu$, and
   $alpha(X_1,Y_1) = d_P(lambda,mu)$.
   But in my problem, I want to fix the random variable $X$.




3. Why the result may be true: the
   space $L^0(Omega, {cal B}, lambda)$ is huge. There
   are lots of random variables with law $mu$. I can't think of any
   obstruction to finding such a random variable.




4. Why the result may be false: the
   space $L^0(Omega, {cal B}, lambda)$ is huge. A compactness
   argument seems hopeless to me. I can't think of any
   construction for finding such a random variable.