The setting is comparing two independent groups represented by random variables $X$, and $Y$. The common language effect size / probability of superiority / relative effect ... is defined as $p(X,Y)=P(X<Y)+0.5P(X=Y)$.
I am wondering what happens if we assume that $X$ and $Y$ are generated from latent variables such that
$$X=X_L + \epsilon_X, Y=Y_L + \epsilon_Y$$
with all random variables being i.i.d., $X_L$, $Y_L$ following arbitrary distributions, $\epsilon_X, \epsilon_Y$ representing error, lets say $E(\epsilon_X)=E(\epsilon_Y)=0$ and both distributions are symmetric.
Specifically, I wonder under what conditions $p(X_L,Y_L)\geq p(X,Y)$ holds. This should be the case if all distributions are normal. In this case, $p(X,Y)$ is a monotonic function of Cohen's d (see Two methods of converting Cohen's d to CL (common language effect size / Probability of Superiority) yield different results?). Adding normally distributed errors to $X_L$ and $Y_L$ leads to increasing the variance but does not change the mean difference. Thus Cohen's d comparing $X$ and $Y$ is smaller than for $X_L$ and $Y_L$. Thus in this case, $p(X_L,Y_L)\geq p(X,Y)$ holds.
I tried other conditions in R and could not yet produce a counter-example. Thus might it even always hold?
