Uniform degree-sensitive averaged exponential-sum conjecture

Let $J$, $(r_i)_{i\in J}$, the polynomials $f_{ij}$, the ideal $\mathcal{I}$, $r$, and $\sigma_0((f_{ij}))$ be as in the degree-sensitive averaged exponential-sum conjecture. Uniform degree-sensitive conjecture. There exist an integer $M$, a positive constant $c$, and positive constants $\{c_L\}_{L\in\widetilde{\mathcal{L}}_{K,1}}$ with $c_L=c$ for $L\in\widetilde{\mathcal{L}}_{K,M}$ such that

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

for every $m\geq1$ and every $L\in\widetilde{\mathcal{L}}_{K,1}$.

This is the stronger version proposed for $Z=\mathbb{A}^n_{\mathcal{O}_K}$, extending the preceding conjecture from $m\geq2$ to $m\geq1$. The source presents it as expected rather than established.