Demeter's continuum small-cap decoupling conjecture for the moment curve in \mathbb{R}^4

Let $11\le p\le 12$. Assume $a\ge b\ge 0$ and $a+b=\frac{p}{2}-3$. Let

$$\Omega=[0,N]\times[0,N^2]\times[0,N^a]\times[0,N^b]$$

and let $\Phi(t)=(t,t^2,t^3,t^4)$ be the moment curve in $\mathbb{R}^4$. Here $n\sim N$ denotes summation over integers comparable to $N$, $x\cdot\Phi(n/N)$ is the Euclidean dot product, and $|\Omega|$ is the volume of $\Omega$. Demeter's conjecture. One should have

$$\int_\Omega\left|\sum_{n\sim N}e\left(x\cdot\Phi\left(\frac{n}{N}\right)\right)\right|^p\,dx\lessapprox N^{\frac p2}|\Omega|,$$

where $e(x)=e^{2\pi i x}$. The paper states that this conjecture is being attacked and its abstract says that it verifies the $p=12$ case, so the status of the full continuum is not established by the supplied material.