The product conjecture for finite transitive sets

Let $X\subset\mathbb R^m$ be a finite transitive set. A finite set $Y\subset\mathbb R^d$ is $k$-Ramsey for $X$ if every $k$-colouring of $Y$ contains a monochromatic subset congruent to $X$. For a positive integer $n$, regard $X^n$ as a subset of $\mathbb R^{mn}$, and call a scalar multiple of it a scaling of $X^n$.

Product conjecture. For every positive integer $k$, there exists a positive integer $n$ such that some scaling of $X^n$ is $k$-Ramsey for $X$.

This is a stronger formulation of the transitive-set characterization: it would immediately imply the “if” direction. It abstracts the product-and-scaling constructions used in known proofs that particular sets are Ramsey. The paper states that it is equivalent to the main transitive-set conjecture, but does not establish it.