Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$.

Question. What is the maximum value of the expectation $$ \mathbb E[\vert f(X_1, \ldots , X_N) \vert] = \int_{\mathbb [0,1]^N} \vert f(x_1, \ldots , x_N) \vert d\mu(x_1) \ldots d\mu(x_N) $$ among all probability measures $\mu$ on $[0,1]$?

This question arises from this post and from this recent answer of Sangchul Lee which seem however tailored for specific functions $f$ and particularly for the case $N=2$.

I am very interested in the case $N\ge 3$; the function $f$ maybe be as smooth as needed (e.g. a polynomial). I have troubles in extending the (very elegant) variational approach of Sangchul Lee's to more variables, as no "bilinear form" is available.

If I am not mistaken, if one considers only a.c. measures then the question becomes: maximize $$ \int_{[0,1]^N} |f(x_1, \ldots, x_N)| g(x_1) g(x_2) \ldots g(x_N) dx_1 dx_2 \ldots dx_N, $$ among functions $g \ge 0$ such that $\int_0^1 g(s)\, ds = 1$. Can this be handled by the general Holder's inequality?

Disclaimer: I have asked this also on MSE but I have not found any answer yet.