Periodic restriction conjecture for the Schrödinger evolution

Let

$$u(x,t)=\sum_{m\in\mathbb{Z}^d}a_m e^{2\pi i(m\cdot x+|m|^2t)},$$

where $a_m$ is supported in

$$Q_N=\{m\in\mathbb{Z}^d:|m_j|\leq N\text{ for all }j\}.$$

For $\lambda>0$, define the superlevel set $U_\lambda(\nu)=\{(x,t):|\nu(x,t)|\geq\lambda\}$. Periodic restriction conjecture. If $|a_m|\leq 1$ for $m\in Q_N$, then

$$|U_\lambda(\nu)\cap[0,1]^{d+1}|\leq C(d,\epsilon)N^{d+2+\epsilon}\lambda^{-\frac{2(d+2)}{d}}$$

for $N^{d/2}\leq\lambda\leq N^d$. This is a periodic analogue of the restriction conjecture, incorporating concentration near rational points on the torus. The source does not state a resolution status.