Multidimensional polynomial Szemerédi property and joint intersectivity

Let $p_1,\ldots,p_r:\mathbb Z^m\to\mathbb Z^k$ be polynomial mappings. For a set $E\subseteq\mathbb Z^k$ of positive upper Banach density, define

$$N_P(E)=\Bigl\{n\in\mathbb Z^m:\text{for some }a\in\mathbb Z^k,\ \{a,a+p_1(n),\ldots,a+p_r(n)\}\subseteq E\Bigr\}.$$

The family has the SPSZ property if $N_P(E)$ is syndetic in $\mathbb Z^m$ for every such $E$. The mappings are jointly intersective if, for every finite-index subgroup $\Lambda\leq\mathbb Z^k$, there exists $n\in\mathbb Z^m$ such that $p_1(n),\ldots,p_r(n)\in\Lambda$. Multidimensional SPSZ conjecture. The family $\{p_1,\ldots,p_r\}$ has the SPSZ property if and only if the mappings $p_1,\ldots,p_r$ are jointly intersective. This extends the multidimensional polynomial Szemerédi theorem by conjecturing an exact characterization of when the syndetic conclusion holds.