The VIP-system nice recurrence conjecture

Let $\mathcal F$ be the collection of all non-empty finite subsets of $\mathbb N$, let $G$ be a countable abelian group, and let $v:\mathcal F\to G$ be a VIP-system, meaning that there is a $t\in\mathbb N$ such that for every pairwise disjoint $\alpha_0,\ldots,\alpha_t\in\mathcal F$,

$$\sum_{\substack{\beta\subseteq\{0,\ldots,t\}\\ \beta\ne\emptyset}}(-1)^{|\beta|}v\left(\bigcup_{j\in\beta}\alpha_j\right)=0_G.$$

VIP-system recurrence conjecture. The VIP-system $v$ is good for nice recurrence.

The statement is presented as a simplified equivalent form of a conjecture of Bergelson and McCutcheon. Its resolution is not supplied in the provided text.