Periodic Schrödinger maximal-function level set conjecture

Let $N$ be a positive integer, let $(a_n)_{n=1}^N$ satisfy

$$\|a_n\|_{l^2}=1,$$

and let $\lambda$ satisfy $N^{\frac14}\leq\lambda\leq N^{\frac12}$. Periodic Schrödinger maximal-function level set conjecture. The estimate

$$\left|\left\{x\in[0,1]:\sup_t\left|\sum_{n=1}^N a_n e(nx+n^2t)\right|\geq\lambda\right\}\right|\lessapprox\frac{N}{\lambda^4}$$

should hold. This is presented as the conjectured sharp level set estimate, up to the $N^\epsilon$ losses hidden in $\lessapprox$, and is equivalent to the corresponding $L^p$ estimate for $p\leq4$; the paper proves the estimate only in a smaller range of $\lambda$.