Condensation and densification conjecture for Vinogradov mean values

Let ${\mathscr A}\subset {\mathbb Z}$ be finite, let $A=|{\mathscr A}|$, and let $s,k\in{\mathbb N}$. Write $J_{s,k}({\mathscr A})$ for the number of solutions in elements of ${\mathscr A}$ to the corresponding Vinogradov system of equations.

Condensation and densification conjecture. One has

$$J_{s,k}({\mathscr A})\leqslant J_{s,k}(\{1,2,\ldots,A\}.$$

Moreover, for each $\varepsilon>0$,

$$J_{s,k}({\mathscr A})\ll_{\varepsilon,s,k} A^{s+\varepsilon}+A^{2s-k(k+1)/2}.$$

The claim says that, among finite sets of integers of a given cardinality, the initial interval is extremal for the Vinogradov mean value, and that the resulting bound removes the dependence on the diameter of the set. Its status is not determined by the supplied material.