Leader–Russell–Walters' transitive-set conjecture for Euclidean Ramsey sets

A finite set $X\subset\mathbb{R}^d$ is Ramsey if, for every $k$, there exists an integer $n$ such that every $k$-colouring of $\mathbb{R}^n$ contains a monochromatic isometric copy of $X$. A finite set is transitive if a group acts transitively on it; a set is isometric to a subset of a finite transitive set when some finite transitive Euclidean set contains an isometric copy of it. Leader–Russell–Walters' conjecture. A set $X\subset\mathbb{R}^d$ is Ramsey if and only if it is isometric to a subset of a finite transitive set. This conjecture is presented as a rival to Graham's conjecture and concerns whether the known Ramsey constructions account for all Ramsey sets. The paper investigates the converse relationship with subsolubility and reports that only two possible exceptions remain among nearly all known examples.