Let $p$ be a sufficiently large prime number and $f\colon\mathbb{F}_{p}\to\mathbb{R}_{\geq 0}$ be a function bounded by 1 such that the average of $f$ over the finite field $\mathbb{F}_{p}$ is at least $\delta>0$. Let $a_{1},\dots,a_{k}\in \mathbb{F}_{p}\backslash\{0\}$ be distinct. I wish to consider the lower bound of the following nonconventional ergodic average $$\Lambda_{a_{1},\dots,a_{k}}(f):=\frac{1}{p^{2}}\sum_{m,n\in\mathbb{F}_{p}}f(m)f(m+a_{1}n)\dots(m+a_{k}n).$$

I know that $\Lambda_{a_{1},\dots,a_{k}}(f)$ is bounded below by a constant $\epsilon$ which depends on $a_{1},\dots,a_{k}, \delta$ (and is independent of $p$ and $f$). My question is: is this lower bound $\epsilon$ dependent only on $k$ and $\delta$ (i.e. independent of the choice of $a_{1},\dots,a_{k}$)?

Here is what I know about this question:

(1) Using the structure theorem (see for example Theorem 3.4 of https://arxiv.org/abs/1212.3681), it suffices to consider the case when $f$ is a periodic nilsequence;

(2) Although I am not aware of a reference, I think the answer to my question for the case $k=2$ should be positive by using some Fourier analysis approach (but what I am most interested in is the case $k>2$).