Equivalence of partition regularity over polynomial and integer patterns

Let $T$ denote the polynomial setting used in the paper, and let $\mathbb{N}$ denote the positive integers. A polynomial pattern $\mathcal{F}$ is partition regular over a domain if every finite coloring of that domain contains a monochromatic realization of $\mathcal{F}$. Polynomial–integer equivalence conjecture. A polynomial pattern $\mathcal{F}$ is partition regular over $T$ if and only if it is partition regular over $\mathbb{N}$. The paper has already shown that partition regularity over $T$ implies partition regularity over $\mathbb{N}$ with primitive recursive bounds, while the converse is posed as an open conjectural strengthening.