Cordoba's Fourier-series norm equivalence conjecture

Let $k$ be a positive integer and $(a_n)_{n\in\mathbb Z}$ a sequence of complex numbers. Define the Fourier series $S(t)$ by

$$S(t) \sim \sum_n a_n e^{2\pi i n^k \theta}.$$

Cordoba's conjecture. For $\alpha<2k$, there exists a constant $C_\alpha$ such that

\left\\|\sum_{n\in\mathbb Z}a_n e^{2\pi i n^k \theta}\right\\|_\alpha \leq C_\alpha \left(\sum_{n\in\mathbb Z}|a_n|^2\right)^{\frac12}.

This asserts equivalence of the $L^2$ and $L^\alpha$ norms for Fourier series supported on $k$th powers. The paper presents the statement as an old conjecture and subsequently discusses the quadratic case, which is disproved for the relevant range of exponents.