The endpoint fourth-moment conjecture for quadratic Weyl sums

For $N\in\mathbb N$, define the quadratic Weyl sum

$$S_N(x)=\sum_{n=1}^N \frac{e^{2\pi i n^2 x}}{n^{1/2}}.$$

Endpoint fourth-moment conjecture. The conjecture is

$$\lVert S_N\rVert_4\simeq (\log N)^{3/4}.$$

The preceding theorem establishes the bounds $(\log N)^{1/2}\lesssim \lVert S_N\rVert_4\lesssim (\log N)^{3/4}$ at the endpoint, so the conjecture asserts that the upper bound gives the correct order of growth.