Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested in proving the following inequality relating even moments of $X$ with even moments of $Y:$

For non-negative integers $i<j,$ we have $$\mathbb{E}X^{2i}\cdot\mathbb{E}Y^{2j}\leq \mathbb{E}X^{2j}\cdot\mathbb{E}Y^{2i}.$$

Some special cases:

a) If $i=0,$ then this follows from Jensen’s inequality.

b) If $X$ is equiprobable on $\{-1,+1\},$ then this follows from the fact that $|Y|\leq 1$ almost surely.

Intuitively, I expect this to be true because $Y$ is a smoothed version of $X.$

Does the inequality follow easily from some known results? If not, what tools might help me prove it?