Finitary polynomial configuration conjecture over rings of integers

Let $K$ be a degree $d$ number field with ring of integers $O_K\cong\mathbb{Z}^d$, and let $p(x)\in K[x]$ be an $O_K$-valued intersective polynomial. Finitary configuration conjecture. For distinct nonzero $r,s\inO_K$, every $\delta,\varepsilon>0$ admits $N=N(\varepsilon,\delta)\in\mathbb{N}$ such that every set $A\subseteq\{1,\ldots,N\}^d$ with $|A|>\delta N^d$ contains at least $(\delta^3-\varepsilon)N^d$ configurations $\{x,x+rp(n),x+sp(n)\}$ for some $n\ne0$. Moreover, if $s/r\in\mathbb{Q}$, or more generally if no two conjugates of $s/r$ are negatives of each other, then the same conclusion holds with at least $(\delta^4-\varepsilon)N^d$ configurations of the form $\{x,x+rp(n),x+sp(n),x+rp(n)+sp(n)\}$. This is proposed as a finitary analogue of the preceding ergodic and combinatorial recurrence results; the supplied text does not establish either finitary assertion.