Bourgain's conjectured torus eigenfunction bounds

Let $B$ be a bounded region in $\mathbb{R}^r$, let $r\geq 3$, and let $(\cdot,\cdot)$ be the positive-definite integral quadratic form used to define the frequency sphere. For coefficients $(a_\xi)$ and frequency scale $N$, Bourgain's torus eigenfunction conjecture. For every $p>\frac{2r}{r-2}$,

$$\left\|\sum_{\substack{\xi\in\mathbb{Z}^r\\ |\xi|=N}}a_\xi e^{i(\xi,x)}\right\|_{L^p(B,dx)}\lesssim N^{\frac{r-2}{2}-\frac{r}{p}}\|a_\xi\|_{\ell^2(\mathbb{Z}^r)},$$

with an $N^\varepsilon$ loss when $r=3,4$. These bounds are presented as the torus input that would yield the compact-Lie-group eigenfunction conjecture for class functions.