Characterization of flip process rules with the same flip process distributions

Let $\mathcal{R}$ and $\mathcal{R}'$ be rules of the same order $k\in\mathbb{N}$, and let $\operatorname{ORB}_{S_k}(\mathcal{H}_{k}\times\mathcal{H}_{k})$ denote the collection of orbits under the action of $S_k$ on pairs of graphs. For an orbit $B$ in this collection, write $\mathcal{R}_{F',H'}$ and $\mathcal{R}'_{F',H'}$ for the corresponding rule weights.

Flip-process distribution conjecture. The following are equivalent:

1. For all $B\in\operatorname{ORB}_{S_k}(\mathcal{H}_{k}\times\mathcal{H}_{k})$,

$$\sum_{(F',H')\in B}\mathcal{R}_{F',H'}=\sum_{(F',H')\in B}\mathcal{R}'_{F',H'}.$$

2. $\mathcal{R}$ and $\mathcal{R}'$ have the same flip process distributions.

This would give an analogous characterization to the paper's result for graphon trajectories. Together with the lifting procedure relating rules of different orders, a positive answer would fully resolve the question of characterizing flip process rules with the same flip process distributions.