Non-partition-regularity of the four-term pattern over bounded-degree polynomial semirings

For $d\geq 1$, let $P_d$ be the set of polynomials in countably many variables with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A finite pattern is partition regular over $P_d$ if every finite coloring of $P_d$ contains a monochromatic realization of the pattern. Bounded-degree obstruction conjecture. If $1\leq d<\infty$, then

$$\{x,y,x+y,xy\}$$

is not partition regular over $P_d$. The conjecture is known for $d=1$, using the two-coloring that colors reducible polynomials red and irreducible polynomials blue; whether the pattern is non-partition-regular over every finite-degree $P_d$ remains open.