Conjectural discrete moment bound for Weyl sums

Let $k\geq 3$, set

$$s_{0}=\frac{(k-1)(k-2)}{2}+1,$$

and suppose that $|\alpha_{k}-a/q|<q^{-2}$ for coprime integers $(a,q)=1$. Let $S_{az}(\boldsymbol{\alpha})$ denote the Weyl sum used in the paper. Conjectural discrete moment bound. The estimate

$$\sum_{a\leq T}|S_{az}(\boldsymbol{\alpha})|\ll Tx\left(\frac{z}{q}+\frac{z\log(q)}{T}+\frac{1}{x^{k}}+\frac{q\log(q)}{Tx^{k}}\right)^{1/(2s_{0})}(Txz)^{\varepsilon}$$

should hold. This would improve the preceding estimates in the range identified in the paper, by gaining an extra factor associated with averaging over the discrete parameter $h$.