The set-uniform Vinogradov mean value conjecture

Let $A \subseteq \mathbb{Z}$ be a finite set. For $s,k \in \mathbb{N}$, let $J_{s,k}(A)$ denote the number of solutions $(x_1,\ldots,x_{2s})\in A^{2s}$ to

$$\sum_{i=1}^{s}(x_i^j-x_{i+s}^j)=0 \qquad (1\leq j\leq k).$$

Set-uniform Vinogradov mean value conjecture. For every real number $\epsilon>0$,

$$J_{s,k}(A) \ll_{s,k,\epsilon} |A|^{\epsilon}\left(|A|^s+|A|^{2s-k(k+1)/2}\right).$$

This conjecture would replace the known factor $N^{\epsilon}$ in the corresponding bound for sets $A\subseteq[N]$ by the intrinsic factor $|A|^{\epsilon}$. The $k=1$ case is trivial, but even the $k=2$ case remains open according to the source.