Polynomial Bohnenblust--Hille bounds for product of cyclic groups

Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2πij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $χ_α$, let $s(α)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust--Hille constants go