The Ramsey-ordering conjecture for permutation group orbits

Let $G$ be a permutation group on $\Omega$ with no finite orbits. Assume that the orbits on $n$-sets can be Ramsey ordered for every positive integer $n$. For positive integers $m$ and $n$, let $c_\alpha$, with $\alpha\in\mathcal{A}$, be a Ramsey ordering of the orbits on $(m+n)$-sets with $N\ge 2(m+n)$, and let $X_\alpha\subseteq\Omega$ be corresponding sets. Suppose this ordering induces Ramsey orderings $d_\beta$, with $\beta\in\mathcal{B}$, and $e_\gamma$, with $\gamma\in\mathcal{C}$, on the $m$-set and $n$-set orbits, respectively. Define $\beta\vee\gamma$ as the least index of an $(m+n)$-set orbit containing a $d_\beta\cup e_\gamma$ decomposition.

Ramsey-ordering conjecture. There exists such a Ramsey ordering for which, for all $\beta,\beta'\in\mathcal{B}$ and $\gamma,\gamma'\in\mathcal{C}$,

$$\beta\vee\gamma<\beta'\vee\gamma\quad\text{if }\beta<\beta'\quad\text{and}\quad\beta\vee\gamma<\beta\vee\gamma'\quad\text{if }\gamma<\gamma'.$$

This conjecture seeks a compatible ordering of orbit decompositions that is strictly increasing in each component. The paper presents it as the main conjecture in the Ramsey-ordering section; no general resolution is given in the supplied text.