As per suggestion, I have decided to post the following as a new question, but it is a followup to this one: Comparison of Rademacher and Gaussian moments under linear transformations

Let $X$ be an $n$ dimensional standard Gaussian and let $X$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z= U^\top X$ is also distributed as a standard Gaussian in $R^n$. This implies that for any function $f: R \to R_+$ where $E[f(G)] < \infty$ for $G$ being a standard gaussian in $R$, we can easily compute $$ E\left[\prod_{i=1}^n f(Z_i)\right] = (E[f(G)])^n,$$ by independence.

Is there a method for bounding expected values of such functions if $X$ was an $n$ dimensional vector where each coordinate is an independent Rademacher $\pm 1$ random variable. For instance, for $Z= U^\top X$, the coordinates are not independent anymore. Is there a way of comparing $E[\prod_{i=1}^n f(Z_i)]$ with $(E[f(G)])^n$ even in this setting? For instance, is it true that $E[\prod_{i=1}^n f(Z_i)] \le \alpha(n)\cdot (E[f(G)])^n$ where $\alpha(n) \le \mathrm{poly}(n)$?

Also, any references to techniques or universality statements that are applicable in such settings will be much appreciated. Thanks!