Thank you very much!

This is very possible, but I simply do not see how.

Thanks! I was hoping for a result of this kind without being able to prove it.

I don't think we have Cov(XZ, Y) = E(D(1-D))V(Y) with my construction. The covariance is positive (and equal to $m_1$ V(Y)) but small enough to ensure the final inequality.

Yes, sorry, I first misread your problem. I have completely modified my answer.

Amazing! I very much doubt I would have found the answer by myself, even though I did look at Latala's paper on Royen's proof. So this was extremely useful :-) (Note: because I am new on this forum, my upvote does not appear yet)