Strict hierarchy conjecture for polynomial partition regularity

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 pattern is partition regular over $P_d$ if every finite coloring of $P_d$ contains a monochromatic realization of the pattern, and it appears in $P_i$ if its one-coloring contains a monochromatic copy of $P_i$. Strict hierarchy conjecture. For all $1\leq i<j$, there exists a pattern $\mathcal{F}$ such that $\mathcal{F}$ is partition regular in $P_j$ but not in $P_i$, while $\mathcal{F}$ appears in $P_i$. This proposes that partition regularity over the bounded-degree polynomial settings has genuinely distinct levels; the paper gives the conjecture as complementary to the non-partition-regularity conjecture for $\{x,y,x+y,xy\}$, and no general resolution is supplied.