The main conjecture for a Vinogradov subsystem

When $k\ge 2$ and $s>0$, we establish that for each $\varepsilon>0$, one has \[ \int_{[0,1)^{k-1}}\biggl| \sum_{1\le x\le X}e(α_2x^2+\ldots +α_k x^k)\biggr|^{2s}\,{\rm d}\boldsymbol α\ll X^{s+\varepsilon}+X^{2s-(k^2+k-2)/2}, \] confirming the main conjecture for a new family of exponential sums beyo