Let $p \ge 1$ be an integer. Does there exist a constant $C_p$ such that for every random variable $X \ge 0$, $$ \mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C_p \mathbb{E} \left[ \left(X^p - \mathbb{E} \left[ X^p \right] \right)^{2} \right] \ \ ? $$