If $\mathbb{N}$ is $2$-coloured then must there exist an infinite set $A=\{a_1<\cdots \}$ such that all expressions of the shape$$\prod _{i\in S_1}{a}_i+\cdots +\prod _{i\in S_k}{a}_i,$$for disjoint $S_1,\dots ,S_k$ (excluding the trivial expressions $a_i$) are the same colour?