Super-modularity preserved by Expectations?

[patrick]

Consider a function $f(a,b)$ that is super-modular in $(a,b)$,
increasing in both variables and convex. Consider a random variable
$\tilde{a}$, We shall denote by $E_a$ the expectations of $f$ wrt to
$\tilde{a}$, assumed to exist.

Is it true that $E_a(f(\tilde{a},b)$ is super-modular in
$(E(\tilde{a},b)$?

Thanks in advance for any hint!