Thank you for your efforts. This was very useful, even if it didn't get us to the sharp bounds - I have awarded the bounty.

Will investigate this, thank you!

This is excellent! I do have a small qualm with the sharpness bit, though: "By determining which value yielded $m$, we can calculate one of the $F$'s." This is not clear to me. If it can be shown that the bounds on the $F$s are sharp, then I agree. Perhaps I am missing something, but it's not immediately obvious to me that $$\max\left\{p_a\int_{0}^{p_{ac}/p_a} Q_a(q)\,dq, p_c\int_{0}^{p_{ac}/p_c} Q_c(q)\,dq\right\} \ \le F_{ac}\ \le \min \left\{ p_a\int_{0}^{p_{ac}/p_a} Q_a(1-q)\,dq, p_c\int_{0}^{p_{ac}/p_c} Q_c(1-q)\,dq\right\}~,$$ for instance, defines a sharp bound.

Yes - even though it is a good bound, it is almost certainly not sharp, since it doesn't use the fact that we observe more than just the moments. We have the entire 2-dimensional marginals here, so I think we should be able to do much better. I appreciate the response, though!