@student You are right, I have added the modulus. Thanks for the interesting connection to the Hausdorff moment problem, I did not know it and I'll certainly look into that.

@PierrePC Interesting random thought! I see your point, you are suggesting that it is enough to solve the problem by consider only symmetric functions, am I right? But I am not following you in your second comment: which "method" are you referring to? D. Hughes' one? Thanks for your interest.

@DavidHughes Mmm, interesting value, I had not found it. Can we use a "double" continuity approach (I mean wrt the measure and to the integrand) to reduce the problem to maximize $E[f(X)]$ where $f$ is an indicator function and formally prove that the max is the one you surmise? Thanks.

@JosiahPark Interesting, thanks. How did you come up with that polynomial? And when is your candidate maximum value attained? Indeed, in the case $N=2$ we have Sangchul Lee's approach yielding that the optimal measure is a.c. with respect to Lebesgue. Can you use your Python code on more general polynomials? Thank you.

@NateEldredge Thank you for sharing your interesting "random thought" :-). However, is it really easier to restrict only to a.c. measures (this amounts to consider the functional I have written at the hand of my post)? Of course a great advantage is that we have now "two" functions $f,g$: is it possible to use Holder/other inequalities to find the maximum? I do not know. Concerning $f$, good point, let us replace $f$ with a polynomial $\tilde{f}$ (uniformly close to $f$).

To the down-voter: could you please suggest what is going wrong in the question so that I can accordingly modify it? Thanks!

Don't even say it as a joke :-)! I do find your contributions really interesting and I thank you for your activity on MO. (Btw thanks also for the answer concerning estimates of incremental quotients in Sobolev spaces: I need some time to digest it :-)). Looking forward to reading you again soon ;-)

I found Tilli's paper extremely interesting, thanks for pointing it out! Accidentally do you know if the Problem 1.3 (mentioned there in the paper) has a solution now? (I have posted the question here in a separate topic). Thanks again for the interesting reference!