Given some (integrable) function $f(x,y)$ with $\partial f/\partial x \leq 0$ and $\partial f/\partial y \leq 0$, and given two random variables $X$ and $Y$ with $\text{cov}(X,Y) \geq 0$, is it true that $$ \mathbb{E}\left( f(x,Y) \mid X = x \right) $$ is weakly decreasing in $x$?
Intuitively, it looks as though it should be but I can't see how to prove it. If it's not, it seems that there should be some minor modification that would make some similar statement true - that would also be very helpful to know.