Parametric Gröbner-basis conjecture for the relation set

Let $P_1,\ldots,P_n\in\mathbb{R}[u]$ be polynomials that map $\mathbb{Z}$ to $\mathbb{Z}$ and are eventually positive. For sufficiently large $t$, define

$$S(t)=\{\mathbf{v}\in\mathbb{Z}^n\mid \mathbf{v}\cdot(P_1(t),\ldots,P_n(t))=0\}.$$

For a finite set $S$ of positive integers, let $G(S)$ be the finite set defined in the source from the lexicographic order, positive and negative parts, and the relation lattice; write $\mathbf{v}^+$ for the positive part of $\mathbf{v}$. A PILP is a finite disjunction of finite conjunctions of parametric inequalities, interpreted through its lattice point set.

Parametric relation-set conjecture. There exists a PILP whose lattice point set equals $G(S(t))$ for $t\gg0$. The same is true for

$$\{\mathbf{v}^+\mid\mathbf{v}\in G(S(t))\}.$$

This proposes a parametric description of the reduced Gröbner-basis-type relation set associated with the polynomially varying integers $P_i(t)$. The supplied text gives no resolution or further evidence of status.