Mixed-norm Strichartz conjecture for integral quadratic forms

Let $(\cdot,\cdot)$ be a positive-definite quadratic form of integral coefficients in $r$ variables, let $B$ be a bounded region in $\mathbb{R}^r$, and let $I$ be a bounded interval. For coefficients $(a_\xi)_{\xi\in\mathbb{Z}^r}$ and $N\geq 1$, the mixed-norm Strichartz conjecture. For all $p,q\geq 2$ satisfying

$$\frac{r}{2}-\frac{2}{p}-\frac{r}{q}>0,$$

one has

$$\left\|\sum_{\substack{\xi\in\mathbb{Z}^r\\ |\xi|\leq N}}a_\xi e^{it(\xi,\xi)+i(\xi,x)}\right\|_{L^p((I,dt),L^q(B,dx))} \lesssim N^{\frac r2-\frac2p-\frac rq}\|a_\xi\|_{\ell^2(\mathbb{Z}^r)}.$$

The conjecture is motivated by the corresponding Euclidean estimates and would imply the optimal compact-Lie-group Strichartz range for class functions. A mixed-Lebesgue decoupling theory needed for the conjecture is described as missing from the literature.