Degree-sensitive averaged exponential-sum conjecture

Let $J\subset\mathbb{N}_{\geq1}$ be finite and non-empty. For each $i\in J$, let $r_i$ be positive, let $f_{ij}\in\mathcal{O}_K[x_1,\dots,x_n]$ be non-constant with $\operatorname{deg}(f_{ij})=i$, and set

$$r=\sum_{i\in J}r_i,\qquad \mathcal{I}=\sum_{i\in J,1\leq j\leq r_i}(f_{ij}).$$

Assume $\mathcal{I}_K\neq(1)$, and let $Z\subset\mathbb{A}^n_{\mathcal{O}_K}$ be a subscheme. Define $\sigma_0((f_{ij}))$ as in the preceding degree-sensitive setup. Degree-sensitive averaged exponential-sum conjecture. There exist a positive constant $c$ and an integer $M$ such that, for all $L\in\widetilde{\mathcal{L}}_{K,M}$ and all $m\geq2$,

$$\left|E_{L,Z,\mathcal{I}}^{(r)}(m)\right|\leq c m^{n+r-1}q_L^{-m\sigma_0((f_{ij}))}.$$

This conjecture strengthens the preceding averaged estimate by replacing the motivic oscillation index with the explicit degree-sensitive threshold. The source studies it, but does not state that it is proved in this generality.