Bergelson–Hindman–Leader partition conjecture for the nonzero rationals

Let $\mathbb{Q}\setminus\{0\}$ be partitioned into finitely many sets,

$$\mathbb{Q}\setminus\{0\}=\bigcup_{i=1}^{r}A_i.$$

For a sequence $\langle x_n\rangle_{n=1}^{\infty}$, let $FS(\langle x_n\rangle_{n=1}^{\infty})$ denote its finite sums and $FP(\langle x_n\rangle_{n=1}^{\infty})$ its finite products. Bergelson–Hindman–Leader conjecture. There exists a finite partition

$$\mathbb{Q}\setminus\{0\}=\bigcup_{i=1}^{r}A_i$$

such that there do not exist $i\in\{1,2,\ldots,r\}$ and a sequence $\langle x_n\rangle_{n=1}^{\infty}$ with

$$FS(\langle x_n\rangle_{n=1}^{\infty})\cup FP(\langle x_n\rangle_{n=1}^{\infty})\subseteq A_i.$$

This extends the corresponding result for dyadic rational numbers; the source presents it as a conjecture from Bergelson, Hindman, and Leader, and its resolution is not indicated here.