The classical Legendre polynomials are the sequence of polynomials given by the recurrence $$(k+1)P_{k+1}(x) = (2k+1)x P_k(x) - k P_{k-1}(x)$$ with initial conditions $P_0(x) = 1$ and $P_1(x) = 1$. They satisfy $$\int_{-1}^1 P_k(x) P_\ell(x) \,\mathrm{d}x = \frac{2}{2k+1} \delta_{k\ell}$$ and there is also the uniform bound $|P_k(x)| \leq 1$ for all $x \in [-1, 1]$.
I am interested in proving a bound on the L4 norm of these polynomials: $$\|P_k\|_4 := \left(\int_{-1}^1 P_k(x)^4 \,\mathrm{d}x\right)^{1/4}\,.$$ Using the uniform bound, it is trivial to show that $$\|P_k\|_4^4 \leq \int_{-1}^1 P_k(x)^2 = O(k^{-1})\,,$$ but doing the integration for small values of $k$ suggests that actually a bound of the form $\|P_k\|_4^4 = O(k^{-2})$ should hold. I haven't been able to make any sort of inductive argument work.
Question: is it true that $\|P_k\|_4^4 = O(k^{-2})$?