Conjecture on popular four-term polynomial configurations over rings of integers

Let $K$ be a number field with ring of integers $O_K$. Let $r,s\thisattacksim$ be distinct and nonzero. For an $O_K$-system $(X,\mathcal{B},\mu,T)$, an $O_K$-valued intersective polynomial $p\in K[x]$, and $A\in\mathcal{B}$, consider the four-term polynomial configuration formed by the transformations $0$, $-rp(n)$, $-sp(n)$, and $-(r+s)p(n)$. Two algebraic numbers are conjugate over $\mathbb{Q}$ when they have the same minimal polynomial over $\mathbb{Q}$. The popular-difference conjecture. For every ergodic measure-preserving $O_K$-system $(X,\mathcal{B},\mu,T)$, every $A\in\mathcal{B}$, every $\varepsilon>0$, and every $O_K$-valued intersective polynomial $p\in K[x]$, the set

$$\left\{n\inO_K:\mu\left(A\cap T^{-rp(n)}A\cap T^{-sp(n)}A\cap T^{-(r+s)p(n)}A\right)>\mu(A)^4-\varepsilon\right\}$$

is syndetic if and only if no two conjugates of $s/r$ over $\mathbb{Q}$ are negatives of each other. The conjecture seeks an algebraic criterion exactly characterizing when the four-term Khintchine-type conclusion holds; the corresponding eigenvalue obstruction is known in related finitary examples, while the asserted equivalence over rings of integers is not established in the supplied text.