My intuition says no, but I couldn't propose a clean counterexample: for example, for any binary r.v. $P[X = a] = p, P[X = b] = 1 - p$ where $a, b > 0$, I found that $$E[X^3] - E[X]E[X^2] = p(1 - p)(a + b)(a - b)^2 > 0,$$ which validates the inequality.
Could anyone give me a counterexample, or prove it (if it is indeed true)?