Hindman's generalized conjecture on finite sums and products

Let $k\in\mathbb{N}$ and let a finite coloring of $\mathbb{N}$ be given. Generalized Hindman's conjecture. There exist infinitely many $x_1<\ldots<x_k\in\mathbb{N}$ such that all finite sums and all finite products formed using distinct elements from $\{x_1,\ldots,x_k\}$ are monochromatic. This extends Hindman's sum-and-product problem from two variables to arbitrarily many selected elements; the supplied status evidence says that this conjecture remains open.