The $n$-dimensional finite-field $(U_k)$-pattern conjecture

Fix a prime $p$ and let $\pi_p:\mathbb{Z}\to\mathbb{F}_p$ be the canonical projection. For $U\subset\mathbb{Z}$, a $U$-pattern in $\mathbb{F}_p^n$ is a set of the form $x+\pi_p(U)\cdot r$ with $x,r\in\mathbb{F}_p^n$ and $r\ne0$. Let $(U_k)$ be a sequence with $|U_k|=k$, and let $g_{n,U_k}(p)$ be the size of the smallest set containing a $U_k$-pattern with basepoint $x$ for every $x\in\mathbb{F}_p^n$. The $n$-dimensional finite-field $(U_k)$-pattern conjecture.

$$\lim_{k\to\infty}\lim_{p\to\infty}\frac{\log g_{n,U_k}(p)}{\log p}=n.$$

This is technically a family indexed by $n$ and by the sequence $(U_k)$. The supplied text does not state any resolution or further consequences.