Even-dimensional monomial distance conjecture over finite p-adic rings

Let $p$ be a sufficiently large prime, let $r$ be a positive integer, and let $E_1,E_2$ be subsets of $(\mathbb{Z}/p^r\mathbb{Z})^n$. For a polynomial

$$F(\mathbf{x})=\sum_{i=1}^n a_i x_i^k$$

with even $n\geq 2$, $k\geq 2$, and $a_i\neq 0$ for all $1\leq i\leq n$, define

$$\Delta_{n,r}(E_1,E_2)=\{F(\mathbf{x}-\mathbf{y}):\mathbf{x}\in E_1,\ \mathbf{y}\in E_2\}$$

and

$$\delta_{E_1,E_2}=\frac{\sqrt{|E_1||E_2|}}{p^{rn}}.$$

Even-dimensional monomial distance conjecture. If $\delta_{E_1,E_2}\gg p^{-n/2}$, then

$$|\Delta_{n,r}(E_1,E_2)|\gg p^r.$$

The conjecture proposes an $r$-uniform density threshold for even-dimensional diagonal monomial distance problems over finite $p$-adic rings. The preceding theorem establishes a related bound for general diagonal polynomials, but its density threshold is independent of $r$ only when the minimum exponent is $2$; the proposed statement remains unproved in the source.