Partition regularity of the reciprocal parallelogram pattern

A finite coloring of $\mathbb{Q}$ is a coloring using finitely many colors. A pattern is partition regular over $\mathbb{Q}$ if every finite coloring of $\mathbb{Q}$ admits parameters making all entries of the pattern the same color.

Reciprocal parallelogram conjecture. The pattern

$$\left\{x,\ x+d,\ x+\frac{1}{d},\ x+d+\frac{1}{d}\right\}$$

is partition regular over $\mathbb{Q}$.

The conjecture asks for a version of a polynomial van der Waerden-type phenomenon allowing negative exponents of $d$. The paper gives no proof and explicitly records the statement as expected to be true.