Discrete Strichartz conjecture for the periodic linear KdV equation

Let $N \ge 1$ be an integer and let $\varepsilon > 0$. Suppose $u_0 \in L^2(\mathbb{T})$ is frequency-localized as

$$u_0(x)=\sum_{n=0}^N a_n e^{2\pi i n x}.$$

Let $u(x,t)$ be the corresponding solution to the periodic linear KdV equation. Discrete Strichartz conjecture. For every $p \ge 2$, one has

$$\|u\|_{L^p(\mathbb{T}^2)} \le C_{p,\varepsilon}N^\varepsilon\left(1+N^{\frac12-\frac4p}\right)\|u_0\|_{L^2(\mathbb{T})},$$

where $C_{p,\varepsilon}$ is a positive constant depending only on $p$ and $\varepsilon$. Equivalently,

$$\left\|\sum_{n=0}^N a_n e^{2\pi i(nx+n^3t)}\right\|_{L^p([0,1]^2)}\lesssim_{p,\varepsilon}N^\varepsilon\left(1+N^{\frac12-\frac4p}\right)\left(\sum_{n=0}^N|a_n|^2\right)^{\frac12}.$$

This is the KdV analogue of the discrete Strichartz estimate for the periodic Schrödinger equation and concerns spacetime bounds for exponential sums along the discrete cubic curve. The supplied text gives no resolution status or supporting evidence, so the conjecture is recorded as open.