Rudin's conjecture that the squares form a $\Lambda(4)$-set

Let $n_1,\ldots,n_k$ be natural numbers, and let $\theta$ range over the unit circle with normalized Lebesgue measure. The squares form a $\Lambda(4)$-set when the fourth moment of every finite exponential sum supported on the squares has the corresponding quadratic bound.

Rudin's conjecture.

$$\int\left|\sum_{i=1}^k\exp(2\pi i n_i^2\theta)\right|^4\,d\theta=O(k^{2+o(1)}).$$

This is a Fourier-analytic conjecture about additive relations among squares and is presented in the source as a longstanding conjecture of Rudin.