Alweiss's non-partition-regularity conjecture for sums and products over polynomial colorings

For $d\in\mathbb{N}$, let $P_d$ be the set of polynomials in a countable set of variables $x_1,x_2,\ldots$ with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A pattern is partition regular over $P_d$ if every finite coloring of $P_d$ contains a monochromatic realization of that pattern.

Alweiss's non-partition-regularity conjecture. There exists $n$ such that, for every $d\in\mathbb{N}$, the pattern

$$\left\{\sum_{i\in S}x_i,\ \prod_{i\in S}x_i: \varnothing\ne S\subseteq[n]\right\}$$

is not partition regular over $P_d$.

The statement is presented as essentially asserting that the methods of the paper are necessary for addressing Hindman's conjecture over polynomial colorings. It remains open; even the weaker assertion that Hindman's conjecture is false over $P_d$ is not known.